3 to varying degrees. Power or exponential equations
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First, let's remember the basic formulas of powers and their properties.
Product of a number a occurs 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 = 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 indicator.
Let us give more examples of exponential equations.
2 x *5=10
16 x - 4 x - 6=0
Now let's look at how exponential equations are solved?
Let's take a simple equation:
2 x = 2 3
This example can be solved even in your head. 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 to formalize this decision:
2 x = 2 3
x = 3
In order to solve such an equation, we removed identical grounds(that is, twos) and wrote down what was left, these are degrees. We got the answer we were looking for.
Now let's summarize our decision.
Algorithm for solving the exponential equation:
1. Need to check the same whether the equation has bases on the right and left. If the reasons are not the same, we are looking for options to solve this example.
2. After the bases become the same, equate degrees and solve the resulting new equation.
Now let's look at a few examples:
Let's start with something 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 is obtained.
x=4 – 2
x=2
Answer: x=2
In the following example you can see that the bases are different: 3 and 9.
3 3x - 9 x+8 = 0
First, move 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 = (3 2) x+8
We get 9 x+8 =(3 2) x+8 =3 2x+16
3 3x = 3 2x+16 Now it is clear that on the left and right sides the bases are the same and equal to three, which means we can discard them and equate the degrees.
3x=2x+16 we get the simplest equation
3x - 2x=16
x=16
Answer: x=16.
Let's look at the following example:
2 2x+4 - 10 4 x = 2 4
First of all, we look at the bases, bases two and four. And we need them to be the same. We transform the four using 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 bother us. What to do with them? If you look closely you can see that on the left side we have 2 2x repeated, 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 entire equation by 6:
Let's imagine 4=2 2:
2 2x = 2 2 bases are the same, we discard them and equate the degrees.
2x = 2 is the simplest equation. Divide it by 2 and 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, you can see that the first three has a degree twice (2x) than the second (just x). In this case, you can solve replacement method. We replace the number with the smallest degree:
Then 3 2x = (3 x) 2 = t 2
We replace all x powers in the equation with t:
t 2 - 12t+27 = 0
We get a quadratic equation. Solving through the discriminant, we get:
D=144-108=36
t 1 = 9
t2 = 3
Returning to the variable x.
Take t 1:
t 1 = 9 = 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 = 3 = 3 x
3 x = 3 1
x 2 = 1
Answer: x 1 = 2; x 2 = 1.
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Degree 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. By multiplying degrees with the same base, their indicators are added:
a m·a n = a m + n .
2. When dividing degrees with the same base, their exponents 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:
(a m) n = a m n .
Each formula above is true 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 a 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 radical number to this power:
4. If you increase the degree of the root in n once and at the same time build into n th power is a radical number, then the value of the root will not change:
5. If you reduce the degree of the root in n extract the root at the same time n-th power of a radical number, then the value of the root will not change:
A degree with a negative exponent. The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power 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 with m< n.
For example. a4:a 7 = a 4 - 7 = a -3.
To formula a m:a n =a m - n became fair when m=n, the presence of zero degree is required.
A degree with a zero index. The power of any number not equal to zero 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 the degree m/n, you need to extract the root n th degree of m-th power of this number A.
REFERENCE MATERIAL ON ALGEBRA FOR GRADES 7-11.
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- Work n factors, each of which is equal A called n-th power of the number A and is designated An.
- The action 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 shows to what power the base is raised is called the exponent. So, An- degree, A– the basis of the 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= a mn
- (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 power (n – natural) number A, not equal to zero, the inverse number is considered n-th power of 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 standard form: a∙10 n, Where 1≤a<10 And n(natural or integer) – is the order of a number written in standard form.
- Expressions that are made up of numbers, variables and their powers using the action of multiplication are called monomials.
- This type of monomial, when the numerical factor (coefficient) comes first, followed by the variables with their powers, is called the standard type of monomial. The sum of the exponents of all variables included in a 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 terms 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 highest of the degrees of its constituent monomials.
- A polynomial of standard form does not contain similar terms and is written in descending order of the degrees of its terms.
- To multiply a monomial by a polynomial, you need 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 the polynomial.
- Taking the common factor out of brackets is the simplest way to factor a polynomial.
- To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of another polynomial and write the resulting products as a sum of monomials. If necessary, add similar terms.
- (a+b) 2 =a 2 +2ab+b 2Square of the sum of two expressions is equal to 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 2Square of the difference of two expressions is equal to 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 is equal to the cube of the first expression plus triple the product of the square of the first expression and the second plus triple the product of the first expression and the square of the second plus the cube of the second expression.
- (a-b) 3 = a 3 -3a 2 b+3ab 2 -b 3Cube of difference of two expressions is equal to 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 =(a-b)(a 2 +ab+b 2) Difference of cubes of two expressions is equal to the product of the difference between the expressions themselves and the partial square of their sum.
- (a+b+c) 2 =a 2 +b 2 +c 2 +2ab+2ac+2bc 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 perfect square of the sum of two expressions: a 2 + 2ab + b 2
Partial square of the sum of two expressions: a 2 + ab + b 2
Function of the form y=x2 called a square function. The graph of a quadratic function is a parabola with its vertex at the origin. Parabola branches y=x² directed upwards.
Function of the form y=x 3 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 located in the 1st and 3rd 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 ordinate axis (Oy). 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 domain of the function value ( -X) is also included in the scope of this function and the equality is satisfied: 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. Equation of the form ax 2 +bx+c=0, Where a, b And c– any real numbers, and a≠0, x– variable, called a quadratic equation.
a– first coefficient, b– second coefficient, c- free member.
Solving incomplete quadratic equations.
- ax 2 =0 – incomplete quadratic equation (b=0, c=0 ). Solution: x=0. Answer: 0.
- ax 2 +bx=0 –incomplete quadratic equation (c=0 ). Solution: x (ax+b)=0 → x 1 =0 or ax+b=0 → x 2 =-b/a. Answer: 0; -b/a.
- ax 2 +c=0 –incomplete quadratic equation (b=0 ); Solution: ax 2 =-c → x 2 =-c/a.
If (-c/a)<0 , then there are no real roots. If (-с/а)>0
- ax 2 +bx+c=0- quadratic equation general view
Discriminant D=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.
- ax 2 +bx+c=0 – quadratic equation private form for even second
Coefficient b
- ax 2 +bx+c=0 – quadratic equation private type provided : a-b+c=0.
The first root is always equal to minus one, and the second root is always equal to minus With, divided by A:
x 1 =-1, x 2 =-c/a.
- ax 2 +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 =1, x 2 =c/a.
Solving the given quadratic equations.
- x 2 +px+q=0 – reduced quadratic equation (the first coefficient is equal to one).
Sum of 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 quadratic equation ax 2 +bx+c=0.
The function of the natural argument is called a number sequence, and the numbers forming the sequence are called terms 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 to the same number for a given sequence d, is called an arithmetic progression. Number d called the difference of an arithmetic progression. In arithmetic progression (a n), i.e. in an arithmetic progression with terms: 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 =a n-1 + d; …
Formula for the nth term of an arithmetic progression.
a n =a 1 +(n-1) d.
Properties of arithmetic progression.
- Each term of an arithmetic progression, starting from the second, is equal to the arithmetic mean of its neighboring terms:
a n =(a n-1 +a n+1):2;
- Each term of an arithmetic progression, starting from the second, is equal to the arithmetic mean of the terms equally spaced from it:
a n =(a n-k +a n+k):2.
Formulas for the sum of the first n terms of an arithmetic progression.
1) S n = (a 1 +a n)∙n/2; 2) S n =(2a 1 +(n-1) d)∙n/2
Geometric progression.
Definition of 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 a given sequence q, is called a geometric progression. Number q called the denominator of a geometric progression. In geometric progression (b n), i.e. in geometric progression b 1, b 2, b 3, b 4, b 5, ..., b n, ... by definition: b 2 = b 1 ∙q; b 3 =b 2 ∙q; b 4 =b 3 ∙q; ... ; b n =b n -1 ∙q.
Formula for the nth term of a geometric progression.
b n =b 1 ∙q n -1 .
Properties of geometric progression.
Formula for the sum of the firstn terms of 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 entire number after the decimal point and the number after the decimal point before the period of the fraction, and the denominator consists of “nines” and “zeros”, and there are as many “nines” as there are digits in the period, and as many “zeros” as there are digits after the decimal point before 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.
Cofunctions of angles that complement each other up to 90° are equal.
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;
Formulas for double and triple arguments.
17) sin2α=2sinαcosα; 18) cos2α=cos 2 α-sin 2 α;
19) 1+cos2α=2cos 2 α; 20) 1-cos2α=2sin 2 α
21) sin3α=3sinα-4sin 3 α; 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.
Evenness (oddness) of 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 by 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 the degree measure of an angle to the radian measure.
3) Converting radian angle measure to degree measure.
Reduction formulas.
Mnemonic rule:
1. Before the reduced function, put the reducible sign.
2. If the argument π/2 (90°) is written an odd number of times, then the function is changed to a cofunction.
Inverse trigonometric functions.
The arcsine of a number (arcsin a) is an angle from the interval [-π/2; π/2 ], whose sine is equal to a.
arcsin(- a)=- arcsina.
The arccosine of a number (arccos a) is an angle from the interval whose cosine is equal to a.
arccos(-a)=π – arccosa.
The arctangent of a number a (arctg a) is an angle from the interval (-π/2; π/2), the tangent of which is equal to a.
arctg(- a)=- arctga.
The arccotangent of a number a (arcctg a) is an angle from the interval (0; π), the cotangent of which is equal to a.
arcctg(-a)=π – arcctg a.
Solving simple 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 =-a, a>0, then t= - 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)
cot t=0, then t = π/2+πn, nϵZ. Solving simple trigonometric inequalities.
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