Some Rules for Exponents

There are a number of rules for exponents but we are only going to look at a few, using base 10, in this module.

Exponent of zero

$10^0$ is defined to be $1$. In fact any number (except $0$) raised to the power zero is $1$.

Negative exponents

The pattern for negative exponents is shown with the examples below. $$10^{-\text{integer}}=\frac{1}{10^{\text{integer}}}$$

Example

\begin{align*} 10^{-4}&=\frac{1}{10^4}\\[1ex] &=\frac{1}{10000}\\[1ex] &= 0.0001\end{align*}

Try these questions. Write each power of $10$ as a fraction (1/number) and as a decimal.

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Multiplying with powers of ten

When you multiply a number by a power of ten with a positive exponent, the result is larger. However, when you multiply a number by a power of ten with a negative exponent, the result is smaller.

Example 1

\begin{align*} 8\times 10^3&=8\times 1000\\ &=8000\end{align*}

Example 2

\begin{align*} 8\times 10^{-3}&=8\times\frac{1}{1000}\\[1ex] &=\frac{8}{1000}\\[1ex] &=8\div 1000\\ &=0.008\end{align*}

Select the correct answer for the meaning of $3.67\times 10^4$

3.67 0.00367 36700 3.670000

Select the correct answer for the meaning of $25.8\times 10^{-3}$

25800 0.0258 0.258 0.000258

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