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

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