Express 108^(1/3) in simplest radical form.
Understand the Problem
The question is asking to express 108^(1/3) in simplest radical form. This involves finding the cube root of 108 and simplifying it to its simplest radical form.
Answer
$3\sqrt[3]{4}$
Answer for screen readers
$3\sqrt[3]{4}$
Steps to Solve
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Convert the fractional exponent to a radical The expression $108^{\frac{1}{3}}$ is equivalent to $\sqrt[3]{108}$.
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Prime factorize the number under the radical The prime factorization of 108 is $2^2 \cdot 3^3$. So we can rewrite the expression as $\sqrt[3]{2^2 \cdot 3^3}$.
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Simplify the radical We can rewrite the radical as $\sqrt[3]{3^3} \cdot \sqrt[3]{2^2}$. Since $\sqrt[3]{3^3} = 3$, the simplified expression is $3\sqrt[3]{2^2}$, or $3\sqrt[3]{4}$.
$3\sqrt[3]{4}$
More Information
The cube root of 108 can be simplified to $3\sqrt[3]{4}$ by factoring out the largest perfect cube from 108, which is 27.
Tips
A common mistake is not fully simplifying the radical. Students might stop after finding a factor that is a perfect cube, but not the largest perfect cube. In this case, they might identify that 108 is divisible by 27 ($3^3$), but then make an arithmetic error. Another common mistake is incorrectly applying the exponent rules or radical rules.
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