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.
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.
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.
\(\displaystyle \left({{7}\over{8}}\right)^{-2} = \mathrm{?}\)
\[\begin{aligned} {\left({{7}\over{8}}\right)^{-2}}&= {{1}\over {\left({{7}\over{8}}\right)^2}}&\blue{\text{by definition}}\\ \\ &={{1}\over {\left({{7}\over{8}}\right)\times\left({{7}\over{8}}\right)}}&\blue{\text{repeated multiplication}}\\ \\&={{1}\over{{{49}\over{64}}}}&\blue{\text{calculation}} \\ \\ &={{64}\over{49}}&\blue{\text{final result}}\end{aligned}\]
Powers with integer exponents
