Power

[chéng fāng]

Mathematical concept

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Find n identical factor product Of operation , called the power, also called the power, power, and the result of the power is called power (power)。 Where, a is called base number (base number), n is called index (exponent)。 When a ⁿ sees The result of a to the nth power , can also be read as "a to the nth power" or "a to the nth power".

A number can be regarded as its own power, and the index 1 is usually omitted. Writing fraction and negative To the nth power of, parentheses should be added. Four operation sequences: first the square, then the brackets (first the parentheses, then the brackets, and finally the braces), followed by multiplication and division, and addition and subtraction at the end.

Calculate the decimal power of a number. If the decimal is a rational number, convert it into a fraction. In particular, the power of 0 of any number other than 0 is equal to 1. A non positive exponential power of 0 has no meaning.

Chinese name: Power
Foreign name: involution
power
Alias: Power 、 Power

Discipline: mathematics
Basic interpretation: Multiplication of multiple identical factors
Expression: aⁿ
Nature: Any number other than 0 to the power of 0 equals 1
Relative operation: Prescription

definition

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Find n identical factor The operation of product is expressed as

。

Note: In the following discussion, the base number is not zero 。

nature

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Power rule of the same base number

Multiply and divide the same base power, the original base is the base, and the sum or difference of the index is the index.^[1]

For example:

1）

；

2）

；

3）

Example of derivation:

set up

In, m=2, n=4, then

Power rule of positive integer exponent

, where

*(i.e. k is positive integer ）

The law of power with exponent of 0

, where

deduction:

Power rule of negative integer exponent

, where

，

deduction:

Power rule of positive fractional index

, where

，

*(i.e. m, n is a positive integer)

Power law of negative fractional exponent

, where,

，

deduction:

When the fractional exponent is raised to a power, when

*, and

The number is meaningless in the real number range

In particular, the power of the non positive exponent of 0 is meaningless^[2]

Square difference

The sum of two numbers multiplied by the difference between two numbers equals their square difference.

Expressed in letters as:

deduction:

Power rule of fraction

prove:

Power rule

Power, base unchanged, exponential multiplication 。

Expressed in letters as:

In particular ：

Power of product

The power of the product. First, each factor in the product is multiplied separately, and then the power obtained is multiplied 。

Expressed in letters as:

The power rule of this product also applies to the power of the product of more than three multipliers. For example:

Same exponential power multiplication

with exponential Multiply, the exponent remains unchanged, and the base number multiplies.

Expressed in letters as:

Perfect square

The square of the sum (or difference) of two numbers is equal to the sum of their squares plus (or minus) twice their product 。

Expressed in letters as:

It is called Complete square formula 。

Cubic difference

Polynomial square

binomial

Isaac Newton Found binomial 。 Binomials are complex operations in the power. In general, the coefficients of the binomial can also be expressed as follows in the order of arrangement:

one

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

…… …… ……

This is the famous Yang Hui Triangle 。

Sign rule of rational number power

(1) The even power of a negative number is positive, and the odd power of a negative number is negative.

(2) Any power of a positive number is a positive number.

(3) Any positive power of 0 is 0.

Quick calculation

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The square of some special numbers can speed up the calculation after mastering the rules. Here is an introduction.

The square of a number consisting of n ones

Observe the following example.

1²=1

11²=121

111²=12321

1111²=1234321

11111²=123454321

111111²=12345654321

……

From the above examples, we can see such a rule; Find the square of the number composed of n ones, first write from 1 to n, and then write from n to 1, that is:

11…1 (n ones) ²=1234…(n-1)n(n-1)…4321

Note: where n only occupies one digit, the full 10 should be carried forward. Of course, such Quick calculation It is not advisable to have too many digits.

The square of a number consisting of n 3s

Specific examples:

3²=9

33²=1089

333²=110889

3333²=11108889

33333²=1111088889

It can be seen that:

33…3 (n 3) ² = 11…11[ (n-1) 1] 0 88…88[ (n-1) 8] nine

One digit is the square of the number of 5

Think of a as the number of 10, so One digit number The square of the number 5 can be written; (10a+5) ². according to Perfect square deduction;

It can be seen that: One digit number is 5 square , equal to the product of the number obtained after removing the single digit number and the number 1 larger than this number, Then write 25.

Pascal language to realize natural number power

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Note: It can only be used to calculate the base number and the exponent are natural numbers, and the power is not greater than 2147483647, otherwise an error will occur

Var a, b, c, i: longint; {The range of longint is large, which is all integers} on [- 21474836482147483647] begin c:=1; {Because all positive integers to the power of 0 are 1} Readln (a, b); {Enter the base number, exponent} If (a=0) and (b=0) then writelin ('Invalid input '); {0 to the power of 0 is meaningless} for i:=1 to b do c:=c*a; {For loop realizes calculation c=a ^ b} writeln(c); {Output c} end.

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