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

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

Square and cubic numbers 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.

\(\displaystyle 10^{-5} = \mathrm{?}\)

\[\begin{aligned} {10^{-5}}&= {{1}\over{10^{5}}}&\blue{\text{by definition}}\\ &=\frac{1}{10\times 10\times 10\times 10\times 10}& \blue{\text{repeated multiplication}} \\ \\ &={{1}\over{100000}}&\blue{\text{final result}}\end{aligned}\]

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