Since ancient times, various peoples of the world have used amulets and talismans for protection. Their...
Abbreviated expression formulas are very often used in practice, so it is advisable to learn all of them by heart. Until this moment, it will serve us faithfully, which we recommend to print and keep in front of our eyes all the time:
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 for a brief multiplication of 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 the name of an expression of the form a 2 −a b + b 2) 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 there is a factorization of a polynomial, FSO is often used in the form with rearranged left and right sides:
The last three identities in the table have their own names. The formula a 2 - b 2 \u003d (a - b) (a + b) is called by the difference of squares formula, a 3 + b 3 \u003d (a + b) (a 2 −a b + b 2) - the formula for the sum of cubes, and a 3 −b 3 \u003d (a − b) (a 2 + a b + b 2) - the difference between cubes... Please note that we did not name the FSU for the corresponding formulas with rearranged parts from the previous table.
Additional formulas
It does not hurt to add a few more identities to the table of formulas for reduced multiplication.
Fields of application of formulas for reduced multiplication (FSU) and examples
The main purpose of the abbreviated multiplication formulas (fsu) is explained by their name, that is, it consists in a brief multiplication of expressions. However, the scope of the FSU is much broader, 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 the implementation of 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.
Decision.
In this expression, squaring can be performed in short, we have 9 y− (1 + 3 y) 2 \u003d 9 y− (1 2 + 2 1 3 y + (3 y) 2)... It remains only to open the brackets and bring similar terms: 9 y− (1 2 + 2 1 3 y + (3 y) 2) \u003d 9 y − 1−6 y − 9 y 2 \u003d 3 y − 1−9 y 2.
Exponentiation is an operation closely related to multiplication; this operation is the result of multiple multiplication of a number by itself. Let's represent by the formula: a1 * a2 *… * an \u003d an.
For example, a \u003d 2, n \u003d 3: 2 * 2 * 2 \u003d 2 ^ 3 \u003d 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 difficult operation. It has to do with multiplication like multiplication and addition. Notation an is a short notation of the n-th number of numbers "a" multiplied by each other.
Consider exponentiation using the simplest examples, moving on to complex ones.
For example, 42.42 \u003d 4 * 4 \u003d 16. Four squared (to the second power) equals sixteen. If you don't understand multiplication 4 * 4, then read our article on multiplication.
Let's look at another example: 5^3. 5^3 = 5 * 5 * 5 = 25 * 5 = 125 ... Five cubed (in the third power) equals one hundred and 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 listed below. There is nothing beyond natural in this, the main thing is to understand the essence and then they will not only be remembered, but also seem easy.
Exponentiation of a monomial
What is a monomial? This is the product of numbers and variables in any quantity. For example, two is a monomial. And this article is about raising to the power of such monomials.
Using the exponentiation formulas, it will not be difficult to calculate the exponentiation of a monomial.
For example, (3x ^ 2y ^ 3) ^ 2 \u003d 3 ^ 2 * x ^ 2 * 2 * y ^ (3 * 2) \u003d 9x ^ 4y ^ 6; If you raise a monomial to a power, then each compound monomial is raised to a power.
Raising to a power a variable that already has a degree, then the degrees are multiplied. For example, (x ^ 2) ^ 3 \u003d x ^ (2 * 3) \u003d x ^ 6;
Negative exponentiation
A negative power is the inverse. What is a reciprocal? Any number X will be inverse 1 / X. That is, X-1 \u003d 1 / X. This is the essence of the negative degree.
Consider an example (3Y) ^ - 3:
(3Y) ^ - 3 \u003d 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 degree. Just isn't it?
Fractional exponentiation
Let's start considering the issue with a specific example. 43/2. What does the 3/2 degree 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 the number (in this case 4).
Then we get the square root of 43 \u003d 2 ^ 3 \u003d 8. Answer: 8.
So, the denominator of a fractional degree can be either 3 or 4 and to infinity any number, and this number determines the degree of the square root extracted from a given number. Of course, the denominator cannot be zero.
Raising a root to a power
If the root is raised to a power equal to the power of the root itself, then the answer will be a radical expression. For example, (√x) 2 \u003d x. And so in any case, the equality of the degree of the root and the degree of erection of the root.
If (√x) ^ 4. Then (√x) ^ 4 \u003d x ^ 2. To check the solution, let's translate 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 \u003d 2. Answer: x \u003d 2.
In any case, the best option is to simply convert the expression to a fractional power expression. If the fraction does not cancel, then this answer will be, provided that the root of the given number is not selected.
Exponentiation of a complex number
What is a complex number? A complex number is an expression having the formula a + b * i; a, b - real numbers. i is the number that, when squared, gives the number -1.
Let's look at an example. (2 + 3i) ^ 2.
(2 + 3i) ^ 2 \u003d 22 +2 * 2 * 3i + (3i) ^ 2 \u003d 4 + 12i ^ -9 \u003d -5 + 12i.
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Online Exponentiation
With our calculator, you can calculate the exponentiation of a number:
Exponentiation grade 7
Exponentiation is only started in the seventh grade.
Exponentiation is an operation closely related to multiplication; this operation is the result of multiple multiplication of a number by itself. Let's represent by the formula: a1 * a2 *… * an \u003d an.
For example, a \u003d 2, n \u003d 3: 2 * 2 * 2 \u003d 2 ^ 3 \u003d 8.
Examples for solution:
Exponentiation presentation
A presentation on raising to a degree for seventh graders. The presentation may clarify some of the confusing points, but it probably won't be thanks to our article.
Outcome
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Abbreviated multiplication formulas.
Study of abbreviated multiplication formulas: the square of the sum and the square of the difference of two expressions; difference of squares of two expressions; sum cube and difference cube of two expressions; sum and difference of cubes of two expressions.
Application of abbreviated multiplication formulas when solving examples.
To simplify expressions, factorize polynomials, and bring polynomials to a 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 the two expressions is square of the first expression plus twice the product of the first expression by the second plus the square of the second expression.
(a + b) 2 \u003d a 2 + 2ab + b 2
2. The squared difference of the two expressions is square of the first expression minus twice the product of the first expression by the second plus the square of the second expression.
(a - b) 2 \u003d a 2 - 2ab + b 2
3. Difference of squarestwo expressions is equal to the product of the difference of these expressions and their sum.
a 2 - b 2 \u003d (a -b) (a + b)
4. Sum cubetwo expressions is equal to the cube of the first expression plus three times the square of the first expression and the second plus three times the first expression and the square of the second plus the cube of the second expression.
(a + b) 3 \u003d a 3 + 3a 2 b + 3ab 2 + b 3
5. Difference cubetwo expressions is equal to the cube of the first expression minus three times 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 - b) 3 \u003d a 3 - 3a 2 b + 3ab 2 - b 3
6. Sum of cubestwo expressions is equal to the product of the sum of the first and second expressions by the incomplete square of the difference of these expressions.
a 3 + b 3 \u003d (a + b) (a 2 - ab + b 2)
7. Difference 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 \u003d (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 \u003d 40 2 + 2 40 1 + 1 2 \u003d 1600 + 80 + 1 \u003d 1681
b) Using the formula for the square of the difference of two expressions, we get
98 2 \u003d (100 - 2) 2 \u003d 100 2 - 2 100 2 + 2 2 \u003d 10000 - 400 + 4 \u003d 9604
Example 2.
Calculate
Using the formula for the difference between the squares of two expressions, we get
Example 3.
Simplify expression
(x - y) 2 + (x + y) 2
We will use the formulas for the square of the sum and the square of the difference of two expressions
(x - y) 2 + (x + y) 2 \u003d x 2 - 2xy + y 2 + x 2 + 2xy + y 2 \u003d 2x 2 + 2y 2
Abbreviated multiplication formulas in one table:
(a + b) 2 \u003d a 2 + 2ab + b 2
(a - b) 2 \u003d a 2 - 2ab + b 2
a 2 - b 2 \u003d (a - b) (a + b)
(a + b) 3 \u003d a 3 + 3a 2 b + 3ab 2 + b 3
(a - b) 3 \u003d a 3 - 3a 2 b + 3ab 2 - b 3
a 3 + b 3 \u003d (a + b) (a 2 - ab + b 2)
a 3 - b 3 \u003d (a - b) (a 2 + ab + b 2)
In the previous lesson, we figured out factoring. We mastered two ways: taking the common factor out of parentheses and grouping. In this tutorial, the next powerful way is: abbreviated multiplication formulas... In short - FSU.
Abbreviated multiplication formulas (square of sum and difference, cube of sum and difference, difference of squares, sum and difference of cubes) are essential in all branches of mathematics. They are used in simplifying expressions, solving equations, multiplying polynomials, canceling fractions, solving integrals, etc. etc. In short, there is every reason to deal with them. Understand where they come from, why they are needed, how to remember them and how to apply them.
Understanding?)
Where do the abbreviated multiplication formulas come from?
Equalities 6 and 7 are not written in a very familiar way. As if the opposite. This is on purpose.) Any equality works both left-to-right and right-to-left. In such a record, it is clearer where the FSO comes from.
They come from multiplication.) For example:
(a + b) 2 \u003d (a + b) (a + b) \u003d a 2 + ab + ba + b 2 \u003d a 2 + 2ab + b 2
That's all, no scientific tricks. We just multiply the parentheses and give similar ones. So it turns out all abbreviated multiplication formulas. Abbreviated multiplication is because in the formulas themselves there is no multiplication of parentheses and a cast of similar ones. Abbreviated.) The result is immediately given.
FSO needs to know by heart. Without the first three, you can not dream of a three, without the rest - of a four and an A.)
Why do we need abbreviated multiplication formulas?
There are two reasons to learn, even to memorize these formulas. The first - a ready-made answer on the machine dramatically reduces the number of errors. But this is not the main reason. But the second ...
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you can get acquainted with functions and derivatives.
Three factors, each of which is x. (\\ displaystyle x.) This arithmetic operation is called "cubing", its result is denoted x 3 (\\ displaystyle x ^ (3)):
x 3 \u003d x ⋅ x ⋅ x (\\ displaystyle x ^ (3) \u003d x \\ cdot x \\ cdot x)For cubing, the reverse operation is cube root extraction. Geometric name of the third degree " cube»Is due to the fact that ancient mathematicians considered the values \u200b\u200bof cubes as cubic numbers, a special kind of curly numbers (see below), since the cube of the number x (\\ displaystyle x) is 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 cubes first n (\\ displaystyle n) positive natural numbers is calculated by the formula:
∑ i \u003d 1 ni 3 \u003d 1 3 + 2 3 + 3 3 +… + n 3 \u003d (n (n + 1) 2) 2 (\\ displaystyle \\ sum _ (i \u003d 1) ^ (n) i ^ (3 ) \u003d 1 ^ (3) + 2 ^ (3) + 3 ^ (3) + \\ ldots + n ^ (3) \u003d \\ left ((\\ frac (n (n + 1)) (2)) \\ right) ^ (2))Formula derivation
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 the reasoning for tables of size n × n.
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The sum of numbers in the k-th (k \u003d 1,2, ...) selected area of \u200b\u200bthe first table:
k 2 + 2 k ∑ l \u003d 1 k - 1 l \u003d k 2 + 2 kk (k - 1) 2 \u003d k 3 (\\ displaystyle k ^ (2) + 2k \\ sum _ (l \u003d 1) ^ (k- 1) l \u003d k ^ (2) + 2k (\\ frac (k (k-1)) (2)) \u003d k ^ (3))And the sum of the numbers in the k-th (k \u003d 1,2, ...) highlighted area of \u200b\u200bthe second table, representing an arithmetic progression:
k ∑ l \u003d 1 n l \u003d k n (n + 1) 2 (\\ displaystyle k \\ sum _ (l \u003d 1) ^ (n) l \u003d 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 \u003d 1 nk 3 \u003d ∑ k \u003d 1 nkn (n + 1) 2 \u003d n (n + 1) 2 ∑ k \u003d 1 nk \u003d (n (n + 1) 2) 2 (\\ displaystyle \\ sum _ (k \u003d 1) ^ (n) k ^ (3) \u003d \\ sum _ (k \u003d 1) ^ (n) k (\\ frac (n (n + 1)) (2)) \u003d (\\ frac (n (n + 1)) (2)) \\ sum _ (k \u003d 1) ^ (n) k \u003d \\ left ((\\ frac (n (n + 1)) (2)) \\ right) ^ (2))Some properties
- In decimal notation, a cube can end with any digit (as opposed to 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 cube digit on the last can be represented in the following table:
Cubes as curly numbers
"Cubic number" Q n \u003d n 3 (\\ displaystyle Q_ (n) \u003d n ^ (3)) historically considered as a kind of spatial curly numbers. It can be represented as the difference of the squares of successive triangular numbers T n (\\ displaystyle T_ (n)):
Q n \u003d (T n) 2 - (T n - 1) 2, n ⩾ 2 (\\ displaystyle Q_ (n) \u003d (T_ (n)) ^ (2) - (T_ (n-1)) ^ (2 ), n \\ geqslant 2) Q 1 + Q 2 + Q 3 + ⋯ + Q n \u003d (T n) 2 (\\ displaystyle Q_ (1) + Q_ (2) + Q_ (3) + \\ dots + Q_ (n) \u003d (T_ (n) ) ^ (2))The difference between two adjacent cubic numbers is the centered hexagonal number.
Expression of a cubic number in terms of tetrahedral Π n (3) (\\ displaystyle \\ Pi _ (n) ^ ((3))).
