3 to varying degrees. Formulas of powers and roots. Properties of an arithmetic progression

REFERENCE MATERIAL ON ALGEBRA FOR GRADES 7-11.

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  • Work n factors, each of which is equal to A called n-th power of a number A and denoted An.
  • The operation by which the product of several equal factors is found is called exponentiation. The number that is raised to a power is called the base of the power. The number that indicates to what power the base is raised is called the exponent. So, An- degree, A- base of degree n- exponent.
  • and 0 =1
  • a 1 = a
  • a m∙ a n= a m + n
  • a m: a n= a m — n
  • (a m) n= amn
  • (a ∙ b) n =a n ∙ b n
  • (a/ b) n= a n/ b n When raising a fraction to a power, both the numerator and denominator of the fraction are raised to that power.
  • (- n) -th degree (n - natural) numbers A, not equal to zero, the number is considered to be the reciprocal of n-th power of a number A, i.e. . a — n=1/ a n. (10 -2 =1/10 2 =1/100=0,01).
  • (a/ b) — n=(b/ a) n
  • The properties of a degree with a natural exponent are also valid for degrees with any exponent.

Very large and very small numbers are usually written in the standard form: a∙10 n, Where 1≤a<10 And n(natural or integer) - is the order of the number written in standard form.

  • Expressions that are made up of numbers, variables and their powers, with the help of multiplication are called monomials.
  • This type of monomial, when the numerical factor (coefficient) is in the first place, followed by the variables with their powers, is called the standard type of monomial. The sum of the exponents of all variables that make up the monomial is called the degree of the monomial.
  • Monomials that have the same letter part are called similar monomials.
  • The sum of monomials is called a polynomial. The monomials that make up a polynomial are called members of the polynomial.
  • A binomial is a polynomial consisting of two terms (monomials).
  • A trinomial is a polynomial consisting of three terms (monomials).
  • The degree of a polynomial is the largest of the degrees of its monomials.
  • The standard form polynomial does not contain such terms and is written in descending order of the powers of its terms.
  • To multiply a monomial by a polynomial, it is necessary to multiply each term of the polynomial by this monomial and add the resulting products.
  • Representing a polynomial as a product of two or more polynomials is called factoring a polynomial.
  • Taking the common factor out of brackets is the simplest way to factorize a polynomial.
  • To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of the other polynomial and write the resulting products as a sum of monomials. If necessary, add like terms.
  • (a+b) 2 =a 2 +2ab+b 2The square of the sum of two expressions equals the square of the first expression plus twice the product of the first expression and the second plus the square of the second expression.
  • (a-b) 2 =a 2 -2ab+b 2The square of the difference of two expressions equals the square of the first expression minus twice the product of the first expression and the second plus the square of the second expression.
  • a 2 -b 2 =(a-b)(a+b) Difference of squares of two expressions is equal to the product of the difference between the expressions themselves and their sum.
  • (a+b) 3 =a 3 +3a 2 b+3ab 2 +b 3Cube of the sum of two expressions equals the cube of the first expression plus three times the square of the first expression times the second plus three times the product of the first expression times the square of the second plus the cube of the second expression.
  • (a-b) 3 = a 3 -3a 2 b+3ab 2 -b 3Difference Cube of Two Expressions equals the cube of the first expression minus three times the product of the square of the first expression and the second plus three times the product of the first expression and the square of the second minus the cube of the second expression.
  • a 3 +b 3 =(a+b)(a 2 -ab+b 2) Sum of Cubes of Two Expressions is equal to the product of the sum of the expressions themselves and the incomplete square of their difference.
  • a 3 -b 3 \u003d (a-b) (a 2 + ab + b 2) Difference of cubes of two expressions is equal to the product of the difference of the expressions themselves and the incomplete square of their sum.
  • (a+b+c) 2 =a 2 +b 2 +c 2 +2ab+2ac+2bc The square of the sum of three expressions is equal to the sum of the squares of these expressions plus all possible doubled pairwise products of the expressions themselves.
  • Reference. The full square of the sum of two expressions: a 2 + 2ab + b 2

Incomplete square of the sum of two expressions: a 2 + ab + b 2

View function y=x2 is called a square function. The graph of a square function is a parabola with vertex at the origin. Parabola branches y=x² directed upward.

View function y=x 3 is called a cubic function. The graph of a cubic function is a cubic parabola passing through the origin. Branches of a cubic parabola y=x³ are in the I and III quarters.

Even function.

Function f is called even if, together with each value of the variable X -X f(- x)= f(x). The graph of an even function is symmetrical about the y-axis. The function y=x 2 is even.

odd function.

Function f is called odd if, together with each value of the variable X from the scope of the function value ( -X) is also included in the scope of this function, and the following equality is true: f(- x)=- f(x) . The graph of an odd function is symmetrical about the origin. The function y=x 3 is odd.

Quadratic equation.

Definition. Type equation ax2+bx+c=0, Where a, b And c are any real numbers, and a≠0, x variable is called a quadratic equation.

a- the first coefficient, b is the second coefficient, c- free member.

Solution of incomplete quadratic equations.

  • ax2=0 – incomplete quadratic equation (b=0, c=0 ). Solution: x=0. Answer: 0.
  • ax2+bx=0 –incomplete quadratic equation (s=0 ). Solution: x (ax+b)=0 → x 1 =0 or ax+b=0 → x 2 =-b/a. Answer: 0; -b/a.
  • ax2+c=0 –incomplete quadratic equation (b=0 ); Solution: ax 2 \u003d -c → x 2 \u003d -c / a.

If (-c/a)<0 , then there are no real roots. If (-s/a)>0

  • ax2+bx+c=0- quadratic equation general view

Discriminant D \u003d b 2 - 4ac.

If D>0, then we have two real roots:

If D=0, then we have a single root (or two equal roots) x=-b/(2a).

If D<0, то действительных корней нет.

  • ax2+bx+c=0 – quadratic equation of particular form for an even second

Coefficient b


  • ax2+bx+c=0 – quadratic equation private type, provided : a-b+c=0.

The first root is always minus one, and the second root is minus With divided by A:

x 1 \u003d -1, x 2 \u003d - c / a.

  • ax2+bx+c=0 – quadratic equation private type, provided: a+b+c=0 .

The first root is always equal to one, and the second root is equal to With divided by A:

x 1 \u003d 1, x 2 \u003d c / a.

Solution of the given quadratic equations.

  • x 2 +px+q=0 – reduced quadratic equation (the first coefficient is equal to one).

The sum of the roots of the reduced quadratic equation x 2 +px+q=0 is equal to the second coefficient, taken with the opposite sign, and the product of the roots is equal to the free term:

ax 2 +bx+c=a (x-x 1)(x-x 2), Where x 1, x 2- roots of the quadratic equation ax2+bx+c=0.

The function of a natural argument is called a numerical sequence, and the numbers that form the sequence are called members of the sequence.

The numerical sequence can be specified in the following ways: verbal, analytical, recurrent, graphic.

A numerical sequence, each member of which, starting from the second, is equal to the previous one, added with the same number for this sequence d is called an arithmetic progression. Number d is called the difference of an arithmetic progression. In arithmetic progression (a n ), i.e. in arithmetic progression with members: a 1 , a 2 , a 3 , a 4 , a 5 , …, a n-1 , a n , … by definition: a 2 = a 1 + d; a 3 = a 2 + d; a 4 = a 3 + d; a 5 = a 4 + d; …; a n \u003d a n-1 + d; …

Formula of the nth member of an arithmetic progression.

a n \u003d a 1 + (n-1) d.

Properties of an arithmetic progression.

  • Each member of the arithmetic progression, starting from the second, is equal to the arithmetic mean of the members adjacent to it:

a n =(a n-1 +a n+1):2;

  • Each member of the arithmetic progression, starting from the second, is equal to the arithmetic mean of the members equally spaced from it:

a n \u003d (a n-k + a n + k): 2.

Formulas for the sum of the first n members of an arithmetic progression.

1) S n = (a 1 +a n)∙n/2; 2) S n \u003d (2a 1 + (n-1) d) ∙ n / 2

Geometric progression.

Definition of a geometric progression.

A numerical sequence, each member of which, starting from the second, is equal to the previous one, multiplied by the same number for this sequence q, is called a geometric progression. Number q called the denominator of a geometric progression. In exponential progression (b n), i.e. exponentially b 1 , b 2 , b 3 , b 4 , b 5 , … , b n , … by definition: b 2 =b 1 ∙q; b 3 \u003d b 2 ∙q; b 4 \u003d b 3 ∙q; … ; b n \u003d b n -1 ∙q.

Formula of the nth member of a geometric progression.

b n \u003d b 1 ∙ q n -1.

Properties of a geometric progression.

The formula for the sum of the firstn terms of a geometric progression.

The sum of an infinitely decreasing geometric progression.

An infinite periodic decimal is equal to a common fraction, in the numerator of which is the difference between the whole number after the decimal point and the number after the decimal point before the fraction period, and the denominator consists of “nines” and “zeros”, moreover, there are as many “nines” as there are digits in the period, and as many “zeroes” as there are digits after the decimal point to the fraction period. Example:

Sine, cosine, tangent and cotangent of an acute angle of a right triangle.

(α+β=90°)

We have: sinβ=cosα; cosβ=sinα; tgβ=ctgα; ctgβ=tgα. Since β=90°-α, then

sin(90°-α)=cosα; cos(90°-α)=sinα;

tg(90°-α)=ctgα; ctg(90°-α)=tgα.

The cofunctions of angles that complement each other up to 90° are equal to each other.

Addition formulas.

9) sin(α+β)=sinα∙cosβ+cosα∙sinβ;

10) sin(α-β)=sinα∙cosβ-cosα∙sinβ;

11) cos(α+β)=cosα∙cosβ-sinα∙sinβ;

12) cos(α-β)=cosα∙cosβ+sinα∙sinβ;

Double and triple argument formulas.

17) sin2α=2sinαcosα; 18) cos2α=cos 2 α-sin 2 α;

19) 1+cos2α=2cos2α; 20) 1-cos2α=2sin 2α

21) sin3α=3sinα-4sin 3α; 22) cos3α=4cos 3 α-3cosα;

Formulas for converting a sum (difference) into a product.

Formulas for converting a product into a sum (difference).

Half argument formulas.

Sine and cosine of any angle.

Even (odd) trigonometric functions.

Of the trigonometric functions, only one is even: y=cosx, the other three are odd, i.e. cos (-α)=cosα;

sin(-α)=-sinα; tg(-α)=-tgα; ctg(-α)=-ctgα.

Signs of trigonometric functions in coordinate quarters.

Values ​​of trigonometric functions of some angles.

Radians.

1) 1 radian is the value of the central angle based on an arc whose length is equal to the radius of the given circle. 1 rad.≈57°.

2) Converting a degree measure of an angle to a radian.

3) Converting the radian measure of an angle to degrees.

Casting formulas.

Mnemonic rule:

1. Before the reduced function put the sign of the reducible.

2. If an odd number of times is taken in the notation of the argument π/2 (90°), then the function is changed to a cofunction.

Inverse trigonometric functions.

The arcsine of the number a (arcsin a) is the angle from the interval [-π/2; π / 2], the sine of which is equal to a.

arc sin(- a)=- arc sina.

The arccosine of the number a (arccos a) is the angle from the interval, the cosine of which is equal to a.

arccos(-a)=π – arccosa.

The arc tangent of the number a (arctg a) is the angle from the interval (-π / 2; π / 2), the tangent of which is a.

arctg(- a)=- arctga.

The arc tangent of the number a (arcctg a) is the angle from the interval (0; π), whose cotangent is equal to a.

arcctg(-a)=π – arcctg a.

Solution of the simplest trigonometric equations.

General formulas.

1) sin t=a, 0

2) sin t = - a, 0

3) cos t=a, 0

4) cos t =-a, 0

5) tg t =a, a>0, then t=arctg a + πn, nϵZ;

6) tg t \u003d -a, a> 0, then t \u003d - arctg a + πn, nϵZ;

7) ctg t=a, a>0, then t=arcctg a + πn, nϵZ;

8) ctg t= -a, a>0, then t=π – arcctg a + πn, nϵZ.

Particular formulas.

1) sin t =0, then t=πn, nϵZ;

2) sin t=1, then t= π/2 +2πn, nϵZ;

3) sin t= -1, then t= - π/2 +2πn, nϵZ;

4) cos t=0, then t= π/2+ πn, nϵZ;

5) cos t=1, then t=2πn, nϵZ;

6) cos t=1, then t=π +2πn, nϵZ;

7) tg t =0, then t = πn, nϵZ;

8) ctg t=0, then t = π/2+πn, nϵZ.

Solution of the simplest trigonometric inequalities.

1) sint

2) sint>a (|a|<1), arcsina+2πn

3) cost

4) cost>a (|a|<1), -arccosa+2πn

5) tgt

6) tgt>a, arctga+πn

7) ctgt

8) ctgt>a, πn

Straight line on the plane.

  • General equation of a straight line: Ax+By+C=0.
  • The equation of a straight line with a slope: y=kx+b (k is the slope).
  • The acute angle between the lines y \u003d k 1 x + b 1 and y \u003d k 2 x + b 2 is determined by the formula:

  • k 1 \u003d k 2 - the condition for parallel lines y \u003d k 1 x + b 1 and y \u003d k 2 x + b 2.
  • The condition of perpendicularity of the same lines:
  • The equation of a straight line having a slope k and passing through

through the point M (x 1; y 1), has the form: y-y 1 \u003d k (x-x 1).

  • The equation of a straight line passing through two given points (x 1; y 1) and (x 2; y 2) has the form:

  • The length of the segment M 1 M 2 with ends at points M 1 (x 1; y 1) and M 2 (x 2; y 2):
  • The coordinates of the point M (x o; y o) - the middle of the segment M 1 M 2

  • The coordinates of the point C (x; y), dividing the segment M 1 M 2 in a given ratio λ between the points M 1 (x 1; y 1) and M 2 (x 2; y 2):

  • Distance from the point M(x o; y o) to the straight line ax+by+c=0:

Circle equation.

  • Circle centered at the origin: x 2 +y 2 =r 2 , r is the radius of the circle.
  • Circle with center at point (a; b) and radius r: (x-a) 2 +(y-b) 2 =r 2 .

Limits.

Transformation (construction) of graphs of functions.

  • Function Graph y=- f(x) is obtained from the graph of the function y=f (x) by mirror reflection from the x-axis.
  • Function Graph y=| f(x)| is obtained by mirror reflection from the abscissa of that part of the graph of the function y \u003d f (x), which lies below the abscissa.
  • Function Graph y= f(| x|) is obtained from the graph of the function y=f (x) as follows: leave part of the graph to the right of the y-axis and display the same part symmetrically to itself with respect to the y-axis.
  • Function Graph y= A∙ f(x) is obtained from the graph of the function y=f (x) by stretching A times along the y-axis. (The ordinate of each point of the graph of the function y \u003d f (x) is multiplied by the number A).
  • Function Graph y= f(k∙ x) obtained from the graph of the function y=f (x) by shrinking k times at k>1 or stretching k times at 0
  • Function Graph y= f(x-m) is obtained from the graph of the function y=f (x) by parallel translation into m unit segments along the x-axis.
  • Function Graph y= f(x)+ n is obtained from the graph of the function y=f (x) by parallel translation into n unit segments along the y-axis.

Periodic function.

  • Function f is called a periodic function with a period Т≠0, if for any x from the domain of definition of the value of this function at points x, T-xAndT+ x are equal, i.e., the equality : f(x)= f(T-x)= f(T+ x)
  • If the function f periodic and has a period T, then the function y= Af(k∙ x+ b), Where A, k And b constant, and k≠0 , is also periodic, and its period is equal to T/| k|.

The limit of the ratio of the increment of a function to the increment of the argument, when the latter tends to zero, is called the derivative of the function at a given point:

  • Function of the form y=a x, where a>0, a≠1, x is any number, are called exponential function.
  • Domain exponential function: D (y)= R - set of all real numbers.
  • Range of values exponential function: E (y)= R+-the set of all positive numbers.
  • Exponential function y=a x increases for a>1.
  • Exponential function y=a x decreases at 0 .

All properties of the power function are valid :

  • and 0 =1 Any number (except zero) to the zero power is equal to one.
  • a 1 = a Any number to the first power is equal to itself.
  • a x∙ay=ax + y When multiplying powers with the same base, the base remains the same, and the exponents are added.
  • a x:ay=ax-y When dividing powers with the same base, the base is left the same, and the exponent of the divisor is subtracted from the exponent of the dividend.
  • (ax) y=axy When raising a power to a power, the base remains the same, and the exponents are multiplied
  • (a∙b)x=ax∙by When raising a product to a power, each of the factors is raised to that power.
  • (a/b)x=ax/by When raising a fraction to a power, both the numerator and denominator of the fraction are raised to that power.
  • a -x \u003d 1 / ax
  • (a/b)-x=(b/a)x.

The logarithm of a number b by reason A (log a b) is the exponent to which the number is to be raised. A to get the number b.

log a b= n, If a n= b. Examples: 1) log 2 8= 3 , because 2 3 =8;

2) log 5 (1/25)= -2 , because 5 -2 \u003d 1/5 2 \u003d 1/25; 3) log 7 1= 0 , since 7 0 =1.

Under the sign of the logarithm can only be positive numbers, moreover, the base of the logarithm is the number a≠1. The value of the logarithm can be any number.

This identity follows from the definition of the logarithm: since the logarithm is an exponent ( n), then by raising the number A, we get the number b.

base logarithm 10 is called the decimal logarithm and when writing, the base 10 and the letter "o" are omitted in the spelling of the word "log".

lg7 = log 10 7, lg7 is the decimal logarithm of the number 7.

base logarithm e(Napier number e≈2.7) is called the natural logarithm.

ln7 = log e 7, ln7 is the natural logarithm of the number 7.

Properties of logarithms valid for logarithms to any base.

log a1=0 The logarithm of unity is zero (a>0, a≠1).

log a a=1 Logarithm of a number A by reason A equals one (a>0, a≠1).

log a (x∙y)=log a x+log a y

The logarithm of the product is equal to the sum of the logarithms of the factors.

log a(x/ y)= log x— log a y

The logarithm of the quotient is equal to the difference between the logarithms of the dividend and the divisor.

log a b=log c b/log c a

Logarithm of a number b by reason A is equal to the logarithm of the number b on a new basis With divided by the logarithm of the old base A on a new basis With.

log a b k= k∙ log a b Degree logarithm ( b k) is equal to the product of the exponent ( k) to the logarithm of the base ( b) of this degree.

log a n b=(1/ n)∙ log a b Logarithm of a number b by reason a n equal to the product of a fraction 1/ n to the logarithm of a number b by reason a.

log a n b k=(k/ n)∙ log a b The formula is a combination of the two previous formulas.

log a r b r = log a b or log a b= log a r b r

The value of the logarithm does not change if the base of the logarithm and the number under the sign of the logarithm are raised to the same power.

  • The function F (x) is called antiderivative for the function f (x) on a given interval, if for all x from this interval F "(x) \u003d f (x).
  • Any antiderivative for the function f (x) on a given interval can be written as F (x) + C, where F (x) is one of the antiderivatives for the function f (x), and С is an arbitrary constant.
  • The set of all antiderivatives F (x) + C of the function f (x) on the interval under consideration is called the indefinite integral and is denoted by ∫f (x) dx, where f (x) is the integrand, f (x) dx is the integrand, x is the variable integration.

1) (∫f(x)dx)"=f(x); 2) d∫f (x) dx=f (x) dx; 3) ∫kf (x) dx=k ∫f (x) dx;

4) ∫dF (x) dx=F (x)+C or ∫F"(x) dx=F (x)+C;

5) ∫(f (x)±g (x)) dx=∫f (x) dx±∫g (x) dx;

6) ∫f (kx+b) dx=(1/k) F (kx+b)+C.

Table of integrals.

The volume of the body of revolution.

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Power formulas used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.

Number c is n-th power of a number a When:

Operations with degrees.

1. Multiplying degrees with the same base, their indicators add up:

a ma n = a m + n .

2. In the division of degrees with the same base, their indicators are subtracted:

3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:

(abc…) n = a n b n c n …

4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:

(a/b) n = a n / b n .

5. Raising a power to a power, the exponents are multiplied:

(am) n = a m n .

Each formula above is correct in the directions from left to right and vice versa.

For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of the ratio is equal to the ratio of the dividend and the divisor of the roots:

3. When raising a root to a power, it is enough to raise the root number to this power:

4. If we increase the degree of the root in n once and at the same time raise to n th power is a root number, then the value of the root will not change:

5. If we decrease the degree of the root in n root at the same time n th degree from the radical number, then the value of the root will not change:

Degree with a negative exponent. The degree of a number with a non-positive (integer) exponent is defined as one divided by the degree of the same number with an exponent equal to the absolute value of the non-positive exponent:

Formula a m:a n = a m - n can be used not only for m> n, but also at m< n.

For example. a4:a 7 = a 4 - 7 = a -3.

To formula a m:a n = a m - n became fair at m=n, you need the presence of the zero degree.

Degree with zero exponent. The power of any non-zero number with a zero exponent is equal to one.

For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.

Degree with a fractional exponent. To raise a real number A to a degree m/n, you need to extract the root n th degree of m th power of this number A.

To the youtube channel of our site site to be aware of all new video lessons.

First, let's recall the basic formulas of degrees and their properties.

Product of a number a happens on itself n times, we can write this expression as a a … a=a n

1. a 0 = 1 (a ≠ 0)

3. a n a m = a n + m

4. (a n) m = a nm

5. a n b n = (ab) n

7. a n / a m \u003d a n - m

Power or exponential equations- these are equations in which the variables are in powers (or exponents), and the base is a number.

Examples of exponential equations:

In this example, the number 6 is the base, it is always at the bottom, and the variable x degree or measure.

Let us give more examples of exponential equations.
2 x *5=10
16x-4x-6=0

Now let's look at how exponential equations are solved?

Let's take a simple equation:

2 x = 2 3

Such an example can be solved even in the mind. It can be seen that x=3. After all, in order for the left and right sides to be equal, you need to put the number 3 instead of x.
Now let's see how this decision should be made:

2 x = 2 3
x = 3

To solve this equation, we removed same grounds(that is, deuces) and wrote down what was left, these are degrees. We got the answer we were looking for.

Now let's summarize our solution.

Algorithm for solving the exponential equation:
1. Need to check the same whether the bases of the equation on the right and on the left. If the grounds are not the same, we are looking for options to solve this example.
2. After the bases are the same, equate degree and solve the resulting new equation.

Now let's solve some examples:

Let's start simple.

The bases on the left and right sides are equal to the number 2, which means we can discard the base and equate their degrees.

x+2=4 The simplest equation has turned out.
x=4 - 2
x=2
Answer: x=2

In the following example, you can see that the bases are different, these are 3 and 9.

3 3x - 9 x + 8 = 0

To begin with, we transfer the nine to the right side, we get:

Now you need to make the same bases. We know that 9=3 2 . Let's use the power formula (a n) m = a nm .

3 3x \u003d (3 2) x + 8

We get 9 x + 8 \u003d (3 2) x + 8 \u003d 3 2 x + 16

3 3x \u003d 3 2x + 16 now it is clear that the bases on the left and right sides are the same and equal to three, which means we can discard them and equate the degrees.

3x=2x+16 got the simplest equation
3x-2x=16
x=16
Answer: x=16.

Let's look at the following example:

2 2x + 4 - 10 4 x \u003d 2 4

First of all, we look at the bases, the bases are different two and four. And we need to be the same. We transform the quadruple according to the formula (a n) m = a nm .

4 x = (2 2) x = 2 2x

And we also use one formula a n a m = a n + m:

2 2x+4 = 2 2x 2 4

Add to the equation:

2 2x 2 4 - 10 2 2x = 24

We gave an example for the same reasons. But other numbers 10 and 24 interfere with us. What to do with them? If you look closely, you can see that on the left side we repeat 2 2x, here is the answer - we can put 2 2x out of brackets:

2 2x (2 4 - 10) = 24

Let's calculate the expression in brackets:

2 4 — 10 = 16 — 10 = 6

We divide the whole equation by 6:

Imagine 4=2 2:

2 2x \u003d 2 2 bases are the same, discard them and equate the degrees.
2x \u003d 2 turned out to be the simplest equation. We divide it by 2, we get
x = 1
Answer: x = 1.

Let's solve the equation:

9 x - 12*3 x +27= 0

Let's transform:
9 x = (3 2) x = 3 2x

We get the equation:
3 2x - 12 3 x +27 = 0

Our bases are the same, equal to three. In this example, it is clear that the first triple has a degree twice (2x) than the second (just x). In this case, you can decide substitution method. The number with the smallest degree is replaced by:

Then 3 2x \u003d (3 x) 2 \u003d t 2

We replace all degrees with x's in the equation with t:

t 2 - 12t + 27 \u003d 0
We get a quadratic equation. We solve through the discriminant, we get:
D=144-108=36
t1 = 9
t2 = 3

Back to Variable x.

We take t 1:
t 1 \u003d 9 \u003d 3 x

That is,

3 x = 9
3 x = 3 2
x 1 = 2

One root was found. We are looking for the second one, from t 2:
t 2 \u003d 3 \u003d 3 x
3 x = 3 1
x 2 = 1
Answer: x 1 \u003d 2; x 2 = 1.

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Degree

Number c (\displaystyle c) called n-th power of a number a (\displaystyle a), If

c = a ⋅ a ⋅ . . . ⋅ a ⏟ n (\displaystyle c=\underbrace (a\cdot a\cdot ...\cdot a) _(n)).

Properties:

  1. (a b) n = a n b n (\displaystyle \left(ab\right)^(n)=a^(n)b^(n))
  2. (a b) n = a n b n (\displaystyle \left((a \over b)\right)^(n)=((a^(n)) \over (b^(n))))
  3. a n a m = a n + m (\displaystyle a^(n)a^(m)=a^(n+m))
  4. a n a m = a n − m (\displaystyle \left.(a^(n) \over (a^(m)))\right.=a^(n-m))
  5. (a n) m = a n m (\displaystyle \left(a^(n)\right)^(m)=a^(nm))
  6. the record does not have the property of associativity (compatibility), that is, in the general case, the left associativity is not equal to the right associativity (a n) m ≠ a (n m) (\displaystyle (a^(n))^(m)\neq a^(\left((n^(m))\right))), the result will depend on the sequence of actions, for example, (2 2) 3 = 4 3 = 64 (\displaystyle (2^(2))^(3)=4^(3)=64), A 2 (2 3) = 2 8 = 256 (\displaystyle 2^(\left((2^(3))\right))=2^(8)=256). It is customary to consider the record a n m (\displaystyle a^(n^(m))) equivalent a (n m) (\displaystyle a^(\left((n^(m))\right))), but instead (a n) m (\displaystyle (a^(n))^(m)) you can just write a n m (\displaystyle a^(nm)) using the previous property. However, some programming languages ​​do not adhere to this convention (see );
  7. exponentiation does not have the property of commutativity  (displacement): generally speaking, a b ≠ b a (\displaystyle a^(b)\neq b^(a)), For example, 2 5 = 32 (\displaystyle 2^(5)=32), But 5 2 = 25 (\displaystyle 5^(2)=25).

real degree

Let a ≥ 0 , r (\displaystyle a\geqslant 0,r) are real numbers, and r (\displaystyle r) is an irrational number. Let's define the value as follows.

As you know, any real number can be approximated, from above and below, by two rational numbers, that is, you can choose for r (\displaystyle r) rational interval [ p , q ] (\displaystyle ) with any degree of accuracy. Then the common part of all corresponding intervals [ a p , a q ] (\displaystyle ) consists of one point, which is taken as a r (\displaystyle a^(r)).

Another approach is based on the theory of series and logarithms (see).

Potentiation

Complex Degree

First, let's show how the exponent is calculated e z (\displaystyle e^(z)), Where e- Euler number, z- arbitrary complex number , z = x + y i (\displaystyle z=x+yi).

e z = e x e y i = e x (cos ⁡ y + i sin ⁡ y) = e x cos ⁡ y + i e x sin ⁡ y . (\displaystyle e^(z)=e^(x)e^(yi)=e^(x)(\cos y+i\sin y)=e^(x)\cos y+ie^(x) \sin y.)

Now consider the general case where a , b (\displaystyle a,b) both are complex numbers. The easiest way to do this is to imagine a (\displaystyle a) in exponential form and using the identity a b = e b Ln ⁡ (a) (\displaystyle a^(b)=e^(b\ \operatorname (Ln) (a))), Where Ln (\displaystyle \operatorname (Ln) )- complex logarithm :

a b = (r e θ i) b = (e Ln ⁡ (r) + θ i) b = e (Ln ⁡ (r) + θ i) b . (\displaystyle a^(b)=(re^((\theta )i))^(b)=(e^(\operatorname (Ln) (r)+(\theta )i))^(b)= e^((\operatorname (Ln) (r)+(\theta )i)b).)

It should be kept in mind that the complex logarithm is a multi-valued function, so that, generally speaking, the complex power is not uniquely defined.

Degree as a function

Because the expression uses two characters ( x (\displaystyle x) And y (\displaystyle y)), then it can be considered as one of three functions:

Useful Formulas

X y = a y log a ⁡ x (\displaystyle x^(y)=a^(y\log _(a)x)) x y = e y ln ⁡ x (\displaystyle x^(y)=e^(y\ln x)) x y = 10 y lg ⁡ x (\displaystyle x^(y)=10^(y\lg x))

The last two formulas are used to raise positive numbers to an arbitrary power on electronic calculators (including computer programs) that do not have a built-in function x y (\displaystyle x^(y)).

Use in oral speech

Recording a n (\displaystyle a^(n)) usually read as " a V n (\displaystyle n) th degree" or " a to the extent n". For example, 10 4 (\displaystyle 10^(4)) reads like "ten to the fourth power" 10 3 / 2 (\displaystyle 10^(3/2)) read as "ten to the power of three second (or: one and a half)."

There are special names for the second and third degrees: squaring and cube, respectively. For example, 10 2 (\displaystyle 10^(2)) reads like "ten squared" 10 3 (\displaystyle 10^(3)) reads like "ten cubed". This terminology originated from ancient Greek mathematics. The ancient Greeks formulated algebraic constructions in the language of geometric algebra (English) Russian. In particular, instead of using the word "multiplication" they were talking about the area a 3 (\displaystyle a^(3)) - this is " a multiplied by itself three times", bearing in mind that three factors are taken a (\displaystyle a). This is not entirely accurate, and can lead to ambiguity, since the number of multiplications will be one less: a 3 = a ⋅ a ⋅ a (\displaystyle a^(3)=a\cdot a\cdot a)(three factors, but two multiplications). Often when they say, "depicted as x I V (\displaystyle x^(IV)) respectively . Beginning with Descartes, the degree was denoted by a "two-story" notation of the form a b (\displaystyle a^(b)).

With the advent of computers and computer programs, the problem arose that in the text of computer programs it is impossible to write the degree in a "two-story" form. In this regard, special icons were invented to indicate the operation of exponentiation. The first such icon was two asterisks.

Some signs of exponentiation in programming languages ​​and computer systems.

A power function is a function of the form y=x n (read as y equals x to the power of n), where n is some given number. Particular cases of power functions are functions of the form y=x, y=x 2 , y=x 3 , y=1/x and many others. Let's talk more about each of them.

Linear function y=x 1 (y=x)

The graph is a straight line passing through the point (0; 0) at an angle of 45 degrees to the positive direction of the Ox axis.

The chart is shown below.

Basic properties of a linear function:

  • The function is increasing and is defined on the whole number axis.
  • It has no maximum and minimum values.

Quadratic function y=x 2

The graph of a quadratic function is a parabola.

Basic properties of a quadratic function:

  • 1. For x=0, y=0, and y>0 for x0
  • 2. The quadratic function reaches its minimum value at its vertex. Ymin at x=0; It should also be noted that the maximum value of the function does not exist.
  • 3. The function decreases on the interval (-∞; 0] and increases on the interval )

Read also

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