Abbreviated expression formulas are very often used in practice, so it is advisable to learn them all by heart. Until this moment, it will serve us faithfully, which we recommend printing out and keeping before your eyes at all times:
The first four formulas from the compiled table of abbreviated multiplication formulas allow you to square and cube the sum or difference of two expressions. The fifth is intended for briefly multiplying the difference and the sum of two expressions. And the sixth and seventh formulas are used to multiply the sum of two expressions a and b by their incomplete square of the difference (this is what an expression of the form a 2 −a b+b 2 is called) and the difference of two expressions a and b by the incomplete square of their sum (a 2 + a·b+b 2 ) respectively.
It is worth noting separately that each equality in the table is an identity. This explains why abbreviated multiplication formulas are also called abbreviated multiplication identities.
When solving examples, especially in which the polynomial is factorized, the FSU is often used in the form with the left and right sides swapped:
The last three identities in the table have their own names. The formula a 2 −b 2 =(a−b)·(a+b) is called difference of squares formula, a 3 +b 3 =(a+b)·(a 2 −a·b+b 2) - sum of cubes formula, A a 3 −b 3 =(a−b)·(a 2 +a·b+b 2) - difference of cubes formula. Please note that we did not name the corresponding formulas with rearranged parts from the previous table.
Additional formulas
It wouldn’t hurt to add a few more identities to the table of abbreviated multiplication formulas.
Areas of application of abbreviated multiplication formulas (FSU) and examples
The main purpose of abbreviated multiplication formulas (fsu) is explained by their name, that is, it consists in briefly multiplying expressions. However, the scope of application of FSU is much wider, and is not limited to short multiplication. Let's list the main directions.
Undoubtedly, the central application of the abbreviated multiplication formula was found in performing identical transformations of expressions. Most often these formulas are used in the process simplifying expressions.
Example.
Simplify the expression 9·y−(1+3·y) 2 .
Solution.
In this expression, squaring can be performed abbreviated, we have 9 y−(1+3 y) 2 =9 y−(1 2 +2 1 3 y+(3 y) 2). All that remains is to open the brackets and bring similar terms: 9 y−(1 2 +2 1 3 y+(3 y) 2)= 9·y−1−6·y−9·y 2 =3·y−1−9·y 2.
Abbreviated multiplication formulas.
Studying abbreviated multiplication formulas: the square of the sum and the square of the difference of two expressions; difference of squares of two expressions; cube of the sum and cube of the difference of two expressions; sums and differences of cubes of two expressions.
Application of abbreviated multiplication formulas when solving examples.
To simplify expressions, factor polynomials, and reduce polynomials to standard form, abbreviated multiplication formulas are used. Abbreviated multiplication formulas need to be known by heart.
Let a, b R. Then:
1. The square 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 2
2. The square 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 - b) 2 = a 2 - 2ab + b 2
3. Difference of squares two expressions is equal to the product of the difference of these expressions and their sum.
a 2 - b 2 = (a -b) (a+b)
4. Cube of sum 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 3
5. Difference cube two expressions is equal to the cube of the first expression minus 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 minus the cube of the second expression.
(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3
6. Sum of cubes two expressions is equal to the product of the sum of the first and second expressions and the incomplete square of the difference of these expressions.
a 3 + b 3 = (a + b) (a 2 - ab + b 2)
7. Difference of cubes two expressions is equal to the product of the difference of the first and second expressions by the incomplete square of the sum of these expressions.
a 3 - b 3 = (a - b) (a 2 + ab + b 2)
Application of abbreviated multiplication formulas when solving examples.
Example 1.
Calculate
a) Using the formula for the square of the sum of two expressions, we have
(40+1) 2 = 40 2 + 2 40 1 + 1 2 = 1600 + 80 + 1 = 1681
b) Using the formula for the square of the difference of two expressions, we obtain
98 2 = (100 – 2) 2 = 100 2 - 2 100 2 + 2 2 = 10000 – 400 + 4 = 9604
Example 2.
Calculate
Using the formula for the difference of the squares of two expressions, we get
Example 3.
Simplify an expression
(x - y) 2 + (x + y) 2
Let's use the formulas for the square of the sum and the square of the difference of two expressions
(x - y) 2 + (x + y) 2 = x 2 - 2xy + y 2 + x 2 + 2xy + y 2 = 2x 2 + 2y 2
Abbreviated multiplication formulas in one table:
(a + b) 2 = a 2 + 2ab + b 2
(a - b) 2 = a 2 - 2ab + b 2
a 2 - b 2 = (a - b) (a+b)
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3
(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3
a 3 + b 3 = (a + b) (a 2 - ab + b 2)
a 3 - b 3 = (a - b) (a 2 + ab + b 2)
Exponentiation is an operation closely related to multiplication; this operation is the result of repeatedly multiplying a number by itself. Let's represent it with the formula: a1 * a2 * … * an = an.
For example, a=2, n=3: 2 * 2 * 2=2^3 = 8 .
In general, exponentiation is often used in various formulas in mathematics and physics. This function has a more scientific purpose than the four main ones: Addition, Subtraction, Multiplication, Division.
Raising a number to a power
Raising a number to a power is not a complicated operation. It is related to multiplication in a similar way to the relationship between multiplication and addition. The notation an is a short notation of the nth number of numbers “a” multiplied by each other.
Consider exponentiation using the simplest examples, moving on to complex ones.
For example, 42. 42 = 4 * 4 = 16. Four squared (to the second power) equals sixteen. If you do not understand multiplication 4 * 4, then read our article about multiplication.
Let's look at another example: 5^3. 5^3 = 5 * 5 * 5 = 25 * 5 = 125 . Five cubed (to the third power) is equal to one hundred twenty-five.
Another example: 9^3. 9^3 = 9 * 9 * 9 = 81 * 9 = 729 . Nine cubed equals seven hundred twenty-nine.
Exponentiation formulas
To correctly raise to a power, you need to remember and know the formulas given below. There is nothing extra natural in this, the main thing is to understand the essence and then they will not only be remembered, but will also seem easy.
Raising a monomial to a power
What is a monomial? This is a product of numbers and variables in any quantity. For example, two is a monomial. And this article is precisely about raising such monomials to powers.
Using the formulas for exponentiation, it will not be difficult to calculate the exponentiation of a monomial.
For example, (3x^2y^3)^2= 3^2 * x^2 * 2 * y^(3 * 2) = 9x^4y^6; If you raise a monomial to a power, then each component of the monomial is raised to a power.
By raising a variable that already has a power to a power, the powers are multiplied. For example, (x^2)^3 = x^(2 * 3) = x^6 ;
Raising to a negative power
A negative power is the reciprocal of a number. What is the reciprocal number? The reciprocal of any number X is 1/X. That is, X-1=1/X. This is the essence of the negative degree.
Consider the example (3Y)^-3:
(3Y)^-3 = 1/(27Y^3).
Why is that? Since there is a minus in the degree, we simply transfer this expression to the denominator, and then raise it to the third power. Simple isn't it?
Raising to a fractional power
Let's start by looking at the issue with a specific example. 43/2. What does degree 3/2 mean? 3 – numerator, means raising a number (in this case 4) to a cube. The number 2 is the denominator; it is the extraction of the second root of a number (in this case, 4).
Then we get the square root of 43 = 2^3 = 8. Answer: 8.
So, the denominator of a fractional power can be either 3 or 4 or up to infinity any number, and this number determines the degree of the square root taken from a given number. Of course, the denominator cannot be zero.
Raising a root to a power
If the root is raised to a degree equal to the degree of the root itself, then the answer will be a radical expression. For example, (√x)2 = x. And so in any case, the degree of the root and the degree of raising the root are equal.
If (√x)^4. Then (√x)^4=x^2. To check the solution, we convert the expression into an expression with a fractional power. Since the root is square, the denominator is 2. And if the root is raised to the fourth power, then the numerator is 4. We get 4/2=2. Answer: x = 2.
In any case, the best option is to simply convert the expression into an expression with a fractional power. If the fraction does not cancel, then this is the answer, provided that the root of the given number is not isolated.
Raising a complex number to the power
What is a complex number? A complex number is an expression that has the formula a + b * i; a, b are real numbers. i is a number that, when squared, gives the number -1.
Let's look at an example. (2 + 3i)^2.
(2 + 3i)^2 = 22 +2 * 2 * 3i +(3i)^2 = 4+12i^-9=-5+12i.
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Exponentiation online
Using our calculator, you can calculate the raising of a number to a power:
Exponentiation 7th grade
Schoolchildren begin raising to a power only in the seventh grade.
Exponentiation is an operation closely related to multiplication; this operation is the result of repeatedly multiplying a number by itself. Let's represent it with the formula: a1 * a2 * … * an=an.
For example, a=2, n=3: 2 * 2 * 2 = 2^3 = 8.
Examples for solution:
Exponentiation presentation
Presentation on raising to powers, designed for seventh graders. The presentation may clarify some unclear points, but these points will probably not be cleared up thanks to our article.
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Abbreviated multiplication formulas or rules are used in arithmetic, more specifically algebra, to speed up the process of evaluating large algebraic expressions. The formulas themselves are derived from rules existing in algebra for multiplying several polynomials.
The use of these formulas provides a fairly quick solution to various mathematical problems, and also helps to simplify expressions. The rules of algebraic transformations allow you to perform some manipulations with expressions, following which you can obtain on the left side of the equality the expression on the right side, or transform the right side of the equality (to obtain the expression on the left side after the equal sign).
It is convenient to know the formulas used for abbreviated multiplication from memory, since they are often used in solving problems and equations. Below are the main formulas included in this list and their names.
Square of the sum
To calculate the square of the sum, you need to find the sum consisting of the square of the first term, twice the product of the first term and the second and the square of the second. In the form of an expression, this rule is written as follows: (a + c)² = a² + 2ac + c².
Squared difference
To calculate the square of the difference, you need to calculate the sum consisting of the square of the first number, twice the product of the first number and the second (taken with the opposite sign) and the square of the second number. In the form of an expression, this rule looks like this: (a - c)² = a² - 2ac + c².
Difference of squares
The formula for the difference of two numbers squared is equal to the product of the sum of these numbers and their difference. In the form of an expression, this rule looks like this: a² - с² = (a + с)·(a - с).
Cube of sum
To calculate the cube of the sum of two terms, you need to calculate the sum consisting of the cube of the first term, triple the product of the square of the first term and the second, triple the product of the first term and the second squared, and the cube of the second term. In the form of an expression, this rule looks like this: (a + c)³ = a³ + 3a²c + 3ac² + c³.
Sum of cubes
According to the formula, it is equal to the product of the sum of these terms and their incomplete squared difference. In the form of an expression, this rule looks like this: a³ + c³ = (a + c)·(a² - ac + c²).
Example. It is necessary to calculate the volume of a figure formed by adding two cubes. Only the sizes of their sides are known.
If the side values are small, then the calculations are simple.
If the lengths of the sides are expressed in cumbersome numbers, then in this case it is easier to use the “Sum of Cubes” formula, which will greatly simplify the calculations.
Difference cube
The expression for the cubic difference sounds like this: as the sum of the third power of the first term, triple the negative product of the square of the first term by the second, triple the product of the first term by the square of the second and the negative cube of the second term. In the form of a mathematical expression, the cube of the difference looks like this: (a - c)³ = a³ - 3a²c + 3ac² - c³.
Difference of cubes
The difference of cubes formula differs from the sum of cubes by only one sign. Thus, the difference of cubes is a formula equal to the product of the difference of these numbers and their incomplete square of the sum. In the form, the difference of cubes looks like this: a 3 - c 3 = (a - c)(a 2 + ac + c 2).
Example. It is necessary to calculate the volume of the figure that will remain after subtracting the yellow volumetric figure, which is also a cube, from the volume of the blue cube. Only the side size of the small and large cube is known.
If the side values are small, then the calculations are quite simple. And if the lengths of the sides are expressed in significant numbers, then it is worth applying the formula entitled “Difference of cubes” (or “Cube of difference”), which will greatly simplify the calculations.
Three factors, each of which is equal x. (\displaystyle x.) This arithmetic operation is called "cube" and its result is denoted x 3 (\displaystyle x^(3)):
x 3 = x ⋅ x ⋅ x (\displaystyle x^(3)=x\cdot x\cdot x)For cubed, the inverse operation is to take the cube root. Geometric name of the third degree " cube"is due to the fact that ancient mathematicians considered the values of cubes as cubic numbers, a special kind of curly numbers (see below), since the cube of the number x (\displaystyle x) equal to the volume of a cube with an edge length equal to x (\displaystyle x).
Sequence of cubes
, , , , , 125, 216, 343, 512, 729, , 1331, , 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97736, 103823, 110592, 117649, 125000, 132651, 140608, 148877, 157464, 166375, 175616, 185193, 195112, 205379, 216000, 226981, 238328…Sum of first cubes n (\displaystyle n) positive natural numbers is calculated by the formula:
∑ i = 1 n i 3 = 1 3 + 2 3 + 3 3 + … + n 3 = (n (n + 1) 2) 2 (\displaystyle \sum _(i=1)^(n)i^(3 )=1^(3)+2^(3)+3^(3)+\ldots +n^(3)=\left((\frac (n(n+1))(2))\right) ^(2))Derivation of the formula
The formula for the sum of cubes can be derived using the multiplication table and the formula for the sum of an arithmetic progression. Considering two 5×5 multiplication tables as an illustration of the method, we will carry out reasoning for tables of size n×n.
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The sum of numbers in the k-th (k=1,2,...) selected area of the first table:
k 2 + 2 k ∑ l = 1 k − 1 l = k 2 + 2 k k (k − 1) 2 = k 3 (\displaystyle k^(2)+2k\sum _(l=1)^(k- 1)l=k^(2)+2k(\frac (k(k-1))(2))=k^(3))And the sum of numbers in the k-th (k=1,2,...) selected area of the second table, representing an arithmetic progression:
k ∑ l = 1 n l = k n (n + 1) 2 (\displaystyle k\sum _(l=1)^(n)l=k(\frac (n(n+1))(2)))Summing over all selected areas of the first table, we get the same number as summing over all selected areas of the second table:
∑ k = 1 n k 3 = ∑ k = 1 n k n (n + 1) 2 = n (n + 1) 2 ∑ k = 1 n k = (n (n + 1) 2) 2 (\displaystyle \sum _(k =1)^(n)k^(3)=\sum _(k=1)^(n)k(\frac (n(n+1))(2))=(\frac (n(n+ 1))(2))\sum _(k=1)^(n)k=\left((\frac (n(n+1))(2))\right)^(2))Some properties
- In decimal notation, a cube can end with any digit (unlike a square)
- In decimal notation, the last two digits of the cube can be 00, 01, 03, 04, 07, 08, 09, 11, 12, 13, 16, 17, 19, 21, 23, 24, 25, 27, 28, 29, 31 , 32, 33, 36, 37, 39, 41, 43, 44, 47, 48, 49, 51, 52, 53, 56, 57, 59, 61, 63, 64, 67, 68, 69, 71, 72 , 73, 75, 76, 77, 79, 81, 83, 84, 87, 88, 89, 91, 92, 93, 96, 97, 99. The dependence of the penultimate digit of the cube on the last one can be presented in the following table:
Cubes as figured numbers
"Cubic number" Q n = n 3 (\displaystyle Q_(n)=n^(3)) historically seen as a type of spatial figured numbers. It can be represented as the difference of the squares of successive triangular numbers T n (\displaystyle T_(n)):
Q n = (T n) 2 − (T n − 1) 2 , n ⩾ 2 (\displaystyle Q_(n)=(T_(n))^(2)-(T_(n-1))^(2 ),n\geqslant 2) Q 1 + Q 2 + Q 3 + ⋯ + Q n = (T n) 2 (\displaystyle Q_(1)+Q_(2)+Q_(3)+\dots +Q_(n)=(T_(n) )^(2))The difference between two adjacent cubic numbers is the centered hexagonal number.
Expressing a cubic number in terms of tetrahedral Π n (3) (\displaystyle \Pi _(n)^((3))).