Step formulas vikoristovuyut in the process of quickness and forgiveness of folding viruses, in the virishennі rіvnіan and irritability.
Number cє n-th step of the number a if:
Operations in steps.
1. Multiplying the steps with the same basis, their indicators are added up:
a ma n = a m + n.
2. At the rozpodіlі stаіnіv z the same basis їх pokanika vіdnіmayutsya:
3. Steps of practice 2 or greater number multipliers for additional steps of these sp_multipliers:
(abc…) n = a n b n c n …
4. The steps of the fraction are more advanced in the introduction of the steps of a given one:
(a/b) n = n/b n .
5. The stars of the steps at the feet, the indicators of the steps are multiplied:
(am) n = a m n .
The skin is shown formula virna u straight ahead zliva to the right and navpak.
For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.
Operations with roots.
1. The root of the creation of many spivmulniki in dobrivnyu dobutku root of these spivmulniki:
2. Root from the root of the root of the root:
3. When the root is added to the rіven, add the zvedi to the whole rіven the root number:
4. How to increase the root steps in n once i at the same hour call in n-th step of the root number, then the value of the root does not change:
5. How to change the root steps in n once and at the same time, pull up the roots n-th step from the root number, then the value of the root does not change:
Step out of a negative indicator. The step of the same number with a non-positive (qіlim) indicator is assigned as one, divided by the step of the same number with the indicator, which is equal to the absolute value of the non-positive indicator:
Formula a m:a n = a m - n you can win not only for m> n, ale i at m< n.
For example. a4: a 7 = a 4 - 7 = a -3.
Schob formula a m:a n = a m - n became fair at m=n, the presence of the zero step is required.
Step out of the zero indicator. Steps of every number, not equal to zero, With a zero indicator, there are more than one.
For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.
Step out of the shotgun display. Schob to call a day number a at the feet m/n it is necessary to win the root n oh world z m th step of th number a.
Enter the number and step, and then press =.
^Step table
Stock: 2 3 = 8
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Level of power - 2 parts
Table of the main steps in algebra in a compact view (picture, handy, easy to explain), to the top of the number, to the side of the step.
DOVIDKOVYY MATERIAL ON ALGEBRI FOR 7-11 CLASS.
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- tvir, dobutok n zmnozhuvachiv a called n-th step of the number a and signify an.
- Diya, from which the tvir is rebuked by a number of equal partners, is called a link at the foot. The number, as it appears at the feet, is called the basis of the step. The number, as it shows, on the foundation of the world, is called the indicator of the step. So, an- Step, a- The basis of the stage, n- step indicator.
- 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 zvedennі at the steps, the shot is made at the whole step and the number and banner of the shot.
- (- n) -th step (n - natural) numbers a, not equal to zero, the number is important, n-th degree of number a, then . a — n=1/ a n. (10 -2 =1/10 2 =1/100=0,01).
- (a/ b) — n=(b/ a) n
- The degree of power with a natural ostentatious is just and for the degrees without being some kind of ostentatious.
Even more large and more small numbers are accepted to be written down in standard look: a∙10 n, de 1≤a<10 і n(Natural or natural) - є the order of the number written in the standard viglyadі.
- Virazi, folded from numbers, changing those steps, with additional multiplication, are called monomials.
- This type of monomial, if there is a numerical multiplier (coefficient) on the first space, and after it change with its steps, is called the standard type of monomial. The sum of the indications of the steps of all the changes that enter the warehouse of the monomial is called the step of the monomial.
- Mononomials that make the same letter part are called similar to monomials.
- The sum of one-members is called a rich-member. The monomials, in that number of foldings, the polynomial, are called the members of the polynomial.
- A binomial is a rich term that consists of two terms (one-members).
- A trinomial is a multi-member, which is composed of three members (one-members).
- The step of a multi-member is the largest of the steps of monomers, which is included up to the new one.
- The rich term of the standard form does not avenge similar terms and entries in the order of the decline in the steps of its terms.
- To multiply a monomial by a polynomial, you need to multiply the monomial of the skin term of a rich term and then create an addition.
- The manifestation of a polynomial, like creating two or more polynomials, is called the decomposition of a polynomial into multipliers.
- The fault of the double multiplier for the bows is the simplest way to lay out the multiplier for multipliers.
- In order to multiply a rich member by a rich member, you need to multiply the skin member of one rich member by the skin member of the other rich member and write down the otrimani create from the sum of the monomers. If necessary, bring similar dodanki.
- (a+b) 2 =a 2 +2ab+b 2Square sumi two viraziv add to the square of the first virase, plus subwinning of the first virase to another, plus the square of the other virase.
- (a-b) 2 =a 2 -2ab+b 2Retail square of two viraziv add to the square of the first virase minus the undersubmission of the first virase to another plus the square of the other virase.
- a 2 -b 2 =(a-b)(a+b) The difference of squares of two verses the cost of restocking the retail of the viruses themselves from their sum.
- (a+b) 3 =a 3 +3a 2 b+3ab 2 +b 3Cube sumi two viraziv add a cube of the first virase plus a third additional square of the first virase to another plus a third additional square of the first virase to the square of another plus a cube of another virase.
- (a-b) 3 = a 3 -3a 2 b+3ab 2 -b 3Retail cube of two viraziv add the cube of the first virase minus the extra square of the first virase to another plus the third extra of the first virase to the square of the other minus the cube of the other virase.
- a 3 +b 3 =(a+b)(a 2 -ab+b 2) The sum of cubes of two viraz dobutka sumi themselves virazіv on the wrong square of their retail.
- a 3 -b 3 \u003d (a-b) (a 2 + ab + b 2) The difference of cubes of two viraziv dobutku raznitsy themselves virazіv on the wrong square of their sum.
- (a+b+c) 2 =a 2 +b 2 +c 2 +2ab+2ac+2bc Square sumi three viraziv add the sum of the squares of these virazis, plus the strengths of the subdivided pairs create the virazis themselves.
- Dovidka. The last square is the sum of two viraziv: a 2 + 2ab + b 2
Non-povny square sum of two viraziv: a 2 + ab + b 2
mind function y=x2 called a square function. The graph of a square function is a parabola with a vertex on the cob of coordinates. Heads of the parabola y=x² upright.
mind function y=x 3 call a cubic function. The graph of a cubic function is a cubic parabola, like passing through the cob of coordinates. Heads of cubic parabola y=x³ found in I and III quarters.
Ready function.
Function f called a steam room, as if at the same time with the skin meanings of the snake X -X f(- x)= f(x). The graph of the paired function is symmetrical along the ordinate axis (Оy). The function y=x2 is a pair.
Unpaired function.
Function f called unpaired, as if at the same time with the skin meanings of the snake X from the area of assigned function value ( -X) also enter the area of designation of function and at which equality is victorious: f(- x)=- f(x) . The graph of an unpaired function is symmetrical to the cob of coordinates. The function y=x3 is unpaired.
Square alignment.
Appointment. Equal to mind ax2+bx+c=0, de a, bі c- be-like real numbers, moreover a≠0, x- Zminna, called square equals.
a- First coefficient, b- Other coefficient, c- Vilniy member.
Razv'yazannya nepovnyh square rіvnyan.
- ax2=0 – not outwardly square alignment (b=0, c=0 ). Solution: x = 0. Response: 0.
- ax2+bx=0 –not outwardly square alignment (Z = 0 ). Solution: x (ax + b) = 0 → x 1 = 0 or ax + b = 0 → x 2 = -b/a. Response: 0; -b/a.
- ax2+c=0 –not outwardly square alignment (b=0 ); Solution: ax 2 = c → x 2 = c/a.
Yakscho (-c/a)<0 , then there are no real roots. Yakscho (-s/a)>0
- ax2+bx+c=0- square alignment infamous looking
Discriminant D \u003d b 2 - 4ac.
Yakscho D>0, then maybe two real roots:
Yakscho D=0, then maybe a single root (or two equal roots) x=-b/(2a).
Yakscho D<0, то действительных корней нет.
- ax2+bx+c=0 – square alignment private view with a double other
Coefficient b
- ax2+bx+c=0 – square alignment private mind : a-b+c=0
The first root is the old root minus one, and the other root is the old minus h, subdivided into a:
x 1 \u003d -1, x 2 \u003d c / a.
- ax2+bx+c=0 – square alignment private mind: a+b+c=0.
The first root is a good one, and the other root is a good one h, subdivided into a:
x 1 \u003d 1, x 2 \u003d c / a.
Rozv'yazannya navigating square lines.
- x 2 +px+q=0 – put square alignment (The first coefficient of the most expensive unit).
The sum of the roots of the induced square alignment x 2 +px+q=0 complementary to another coefficient taken with the opposite sign, and the addition of the root relative to the free member:
ax 2 +bx+c=a (x-x 1)(x-x 2), de x 1, x 2- root of square alignment ax2+bx+c=0.
The function of a natural argument is called a numerical sequence, and the numbers that satisfy the sequence are members of the sequence.
The numerical sequence can be set in the following ways: verbal, analytical, recurrent, graphic.
Numerical sequence, a skin member of a kind, starting from another, older than the front, folded by him for this sequence by a number d called arithmetic progression. Number d called the difference of arithmetic progression. In arithmetic progression (an), then in arithmetic progression with members: a 1 , a 2 , a 3 , a 4 , a 5 , …, a n-1 , a n , … for appointments: 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 n-th member of the arithmetic progression.
a n = 1 + (n-1) d.
The dominance of arithmetic progression.
- The skin member of the arithmetic progression, starting from another, is closer to the arithmetic mean of the sudial member:
an=(an-1+an+1):2;
- The skin member of the arithmetic progression, starting from another, is closer to the arithmetic mean equal of the distant members:
an=(an-k+an+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 \u003d (2a 1 + (n-1) d) ∙ n / 2
geometric progression.
Designated geometric progression.
Numerical sequence, skin member of this, starting from another, older than the previous one, multiplied by the same number for this sequence q, called geometric progression. Number q called the sign of geometric progress. In a geometric progression (b n), then in a geometric progression b 1, b 2, b 3, b 4, b 5, ..., b n, ... for the appointments: 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 n-th member of the geometric progression.
b n \u003d b 1 q n -1.
The dominance of geometric progression.
Formula sumi firstn terms of geometric progression.
The sum of infinitely slow geometric progression.
Unlimited periodic decimal fraction is more expensive than the grand fraction, in the numeral book, there is a difference between the last number after the Komi and the number after the Komi before the fractional period, and the banner is made up of “nine” and “zero”, moreover, “nine” styles, the number of numbers in the period, and “zero” stilks, skіlki digits after the Komi to the fractional period. Butt:
Sine, cosine, tangent and cotangent of the acute cut of a straight-cut tricot.
(α+β=90°)
May: sinβ=cosα; cosβ=sinα; tgβ=ctgα; ctgβ=tgα. Oskilki β=90°-α, then
sin(90°-α)=cosα; cos(90°-α)=sinα;
tg(90°-α)=ctgα; ctg(90°-α)=tgα.
The co-functions of the kutivs, which complement one another up to 90 °, are equal to each other.
Addendum 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 of subvariant and subvariant arguments.
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 sumi (retail) on TV.
Formulas for the transformation of creativity in the bag (retail).
Half Argument Formulas.
The sine is the cosine of whatever kuta.
parity (non-parity) of trigonometric functions.
Of the trigonometric functions, there is more than one pair: y=cosx, three trigonometric functions are unpaired, so cos (-α)=cosα;
sin(-α)=-sinα; tg(-α)=-tgα; ctg(-α)=-ctgα.
Signs of trigonometric functions behind coordinate quarters.
Values of trigonometric functions of deyaky cutivs.
Radiani.
1) 1 radian - the value of the central kuta, which spirals onto the arc, the length of which is equal to the radius of the given stake. 1 rad.≈57°.
2) Conversion of the degree setting of the kuta to the radian.
3) Converting the radian world kuta to degrees.
Guidance formulas.
Mnemonic rule:
1. Before the hovered function, put a sign to hover.
2. If the argument π/2 (90°) is written an unpaired number of times, then the function is changed to a cofunction.
Return trigonometric functions.
The arcsine of the number a (arcsin a) is the cut from the gap [-π/2; π / 2], the sine of which is more expensive a.
arc sin(- a)=- arc sina.
The arccosine of the number a (arccos a) is called the cut from the gap, the cosine of any other a.
arccos(-a)=π - arccosa.
The arc tangent of the number a (arctg a) is the cut from the interval (-π / 2; π / 2), the tangent of which is more expensive a.
arctg(- a)=- arctga.
The arc tangent of the number a (arcctg a) is called the cut from the interval (0; π), the cotangent of any other a.
arcctg(-a)=π - arcctg a.
Verification of the simplest trigonometric equalities.
Zagalnі 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. Private 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. The solution to the simplest trigonometric irregularities.
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