What is Power? — Mathematics

Definition 56.1 Primary & Middle School Mathematics · Chapter 56 — Powers

For a number $a$ and a whole number $n \geq 1$ :

a^n = \underbrace{a \times a \times \dots \times a}_{n \text{ factors}},

read “ $a$ to the (power) $n$ ”; $a$ is the base, $n$ the exponent. Special names: $a^2$ is “ $a$ squared”, $a^3$ “ $a$ cubed”. By convention:

a^1 = a, \qquad a^0 = 1 \ (a \neq 0), \qquad a^{-n} = \frac{1}{a^n}.

Powers of 2: each bar is twice the previous one. Growth by repeated multiplication runs away much faster than growth by repeated addition. — Powers of $2$ : each bar is twice the previous one. Growth by repeated multiplication runs away much faster than growth by repeated addition.

Examples

Example 56.2

$3^4 = 3 \times 3 \times 3 \times 3 = 81$ ; $10^3 = 1000$ ; $5^{-2} = \frac{1}{25} = 0.04$ ; $(-2)^3 = -8$ and $(-2)^4 = +16$ (Example 54.5). Watch out: $a^n$ is not $a \times n$ : $2^5 = 32$ , not $10$ .

Example 56.4

2^3 \times 2^5 = 2^8 = 256, \qquad \frac{7^6}{7^4} = 7^2 = 49, \qquad \left(5^2\right)^3 = 5^6, \qquad 2^4 \times 5^4 = 10^4 .

A trap: the rules apply to a common base (or a common exponent, for the last one). No rule simplifies $2^3 \times 5^2$ — just compute: $8 \times 25 = 200$ .

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Definition 63.7 Primary & Middle School Mathematics · Chapter 63 — Fractions and Powers

For a real number $a$ and a positive integer $n$ , the power $a^n$ is the product of $n$ factors equal to $a$ :

a^n = \underbrace{a \times a \times \dots \times a}_{n \text{ factors}} .

By convention $a^0 = 1$ (for $a \neq 0$ ), and negative exponents denote inverses:

a^{-n} = \frac{1}{a^n}.

Examples

Example 63.13

The Earth–Sun distance is about $149\,600\,000$ km $= 1.496 \times 10^8$ km. The size of a water molecule is about $0.000\,000\,000\,28$ m $= 2.8 \times 10^{-10}$ m. To compute with such numbers, group the decimal parts and the powers of ten:

(3 \times 10^5) \times (2.5 \times 10^{-2}) = (3 \times 2.5) \times 10^{5 + (-2)} = 7.5 \times 10^{3}.

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Definition 30.8 High School Mathematics · Chapter 30 — Matrices and Graphs

For a square matrix $A$ and $k \in \N$ , $A^k = A \times \dots \times A$ ( $k$ factors), with $A^0 = I$ .

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Examples

Examples

Related terms