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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.
Index Form | Fraction Decimal | |
$10^{-3}$ | ||
$10^{-2}$ | ||
$10^{-1}$ |
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*}
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