Example 3
Completion requirements
| Example 3 |
Simplify the radical expression \(\frac{{\sqrt[4]{{64t^3 }}}}{{\sqrt[4]{{4t}}}}\). Identify any restrictions on the variable.
Step 1: State any restrictions on the variable.
Because the index is even (four), \(t^3\) must be greater than or equal to zero. Because \(t\) is found in the denominator, \(t\) cannot equal zero.
\(\begin{align}
t^3 &> 0 \\
t &> 0 \\
\end{align}\)
Step 2: Simplify the expression.
\(\begin{align}
\frac{{\sqrt[4]{{64t^3 }}}}{{\sqrt[4]{{4t}}}} &= \sqrt[4]{{\frac{{64t^3 }}{{4t}}}} \\
&= \sqrt[4]{{16t^2 }} \\
&= 2\sqrt[4]{{t^2 }} \\
&= 2t^{\frac{2}{4}} \\
&= 2t^{\frac{1}{2}} \\
&= 2\sqrt t \\
\end{align}\)
Because the index is even (four), \(t^3\) must be greater than or equal to zero. Because \(t\) is found in the denominator, \(t\) cannot equal zero.
\(\begin{align}
t^3 &> 0 \\
t &> 0 \\
\end{align}\)
Step 2: Simplify the expression.
\(\begin{align}
\frac{{\sqrt[4]{{64t^3 }}}}{{\sqrt[4]{{4t}}}} &= \sqrt[4]{{\frac{{64t^3 }}{{4t}}}} \\
&= \sqrt[4]{{16t^2 }} \\
&= 2\sqrt[4]{{t^2 }} \\
&= 2t^{\frac{2}{4}} \\
&= 2t^{\frac{1}{2}} \\
&= 2\sqrt t \\
\end{align}\)