- Write numbers in exponential form
Writing Numbers in Exponential Form
When solving exponential equations, it’s often helpful to rewrite numbers as powers of the same base. This lets us compare exponents directly.
Some numbers are perfect powers of smaller bases.
Fractions are often powers of whole numbers, but with negative exponents.
[latex]\frac{1}{a^n}=a^{-n}[/latex]
Rewrite [latex]\frac{1}{25}[/latex] in exponential form.
Start by identifying the exponential form of [latex]25[/latex].Since [latex]25=5^2[/latex] we know [latex]\frac{1}{25}=\frac{1}{5^2}[/latex].
Now, rewrite the fraction with a negative exponent.
[latex]\frac{1}{5^2}=5^{-2}[/latex]
Power Property of ExponentsWhen raising a power to another power, multiply the exponents:
[latex](a^m)^n=a^{m*n}[/latex]
Rewrite [latex]25^x[/latex] using base [latex]5[/latex].
\begin{aligned}
25^x &= (5^2)^x && \text{Rewrite 25 as } 5^2 \\[6pt]
&= 5^{2x} && \text{Apply the power property}
\end{aligned}
Rewrite [latex]9^{x+1}[/latex] using base [latex]3[/latex]
\begin{aligned} 9^{x+1} &= (3^2)^{x+1} && \text{Rewrite 9 as } 3^2 \\[6pt] &= 3^{2(x+1)} && \text{Apply the power property} \\[6pt] &= 3^{2x+2} && \text{Distribute the multiplication} \end{aligned}