Powers with integer exponents

Calculating with numbers: Calculating with powers and roots

Powers with integer exponents

In scientific notation, a number is written as the product of a number that consists only of significant digits and a power of 10. In order to be able to calculate with these numbers you must be able work with powers of 10. Because the calculation rules hold for powers with a randomly chosen base we brush up the knowledge about arithmetical work with powers.

Exponentiation with an integer as exponent

For any nonzero number $g$ and any positive integer $k$ one defines $\begin{aligned} g^k&=\blue{ \overbrace{\color{black}{g \times g \times \cdots \times g}}^{k\mathrm{\;times}}} \\[0.2cm] g^0\; & = 1 \\[0.2cm] g^{-k}\; & = \frac{1}{g^k} \end{aligned}$

This way, $g^n$ is defined for each integer $n$ . The number $g$ is called the base and $n$ is called the exponent.

Thus, exponentiation is repeated multiplication.

You pronounce the index (power) of a number, say of the form $2^3$ as "two to the third power" or as "two to the power three".

Examples
$\begin{aligned}2^{-3}&=\frac{1}{2^3}=\frac{1}{2\times 2\times 2}=\frac{1}{8}\\[0.2cm] 2^{-2}&=\frac{1}{2^2}=\frac{1}{2\times 2}=\frac{1}{4}\\[0.2cm] 2^{-1}&=\frac{1}{2^1}=\frac{1}{2}\\[0.2cm] 2^0&=1\\[0.2cm] 2^1&=2\\[0.2cm] 2^2&=2\times 2=4\\[0.2cm] 2^3&=2\times 2\times 2=8\end{aligned}$

In case of exponentiation you must pay attention to a minus sign, whether it belongs to the base or not. Brackets resolve this: $(-3)^2 = (-3)\times(-3)=9\qquad\text{and}\qquad -3^2=-(3\times 3)=-(9)=-9$

A square number, also referred to as a perfect square, is a natural number that can be written as the square of a natural number. For example, $36 = 6^2$ and $121 = 11^2$ are square numbers.

A similar nomenclature is use for the third power of a natural number. A cubic number, also referred to as a perfect cube, is a natural number that can be written as the third power of a natural number. For example, $27 = 3^3$ and $125 = 5^3$ are cubic numbers.

Alternative way of writing To let it all make sense and not to get confused by exponents like $-1$ , $0$ and $1$ , you can always work out an index by first writing a $1$ and then multiply it as many times by the base as the exponent indicates. For a negative exponent you can always work out an index by first writing a $1$ and then multiply it as many times by the reciprocal value of the base as the absolute value of the exponent indicates. For example: $\begin{aligned}3^2 &= 1\times 3\times 3\\[0.2cm] 3^1 &= 1\times 3\\[0.2cm] 3^0 &= 1\\[0.2cm] 3^{-1} &= 1\times \frac{1}{3}\\[0.2cm] 3^{-2} &= 1\times \frac{1}{3} \times \frac{1}{3}\end{aligned}$

Look at some examples by repeatedly clicking on the new example button in the dynamic example below.

$\displaystyle \left({{3}\over{10}}\right)^3 = \mathrm{?}$

$\begin{aligned} {\left({{3}\over{10}}\right)^3}&= {{3}\over{10}}\times {{3}\over{10}}\times {{3}\over{10}}&\blue{\text{repeated multiplication}} \\ \\&= {{27}\over{1000}}&\blue{\text{final result}}\end{aligned}$

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