To simplify the expression [tex]\( \sqrt[3]{81} \)[/tex], let's follow these steps:
1. Express 81 as a product of prime factors:
[tex]\( 81 = 3 \times 3 \times 3 \times 3 \)[/tex]
This can be written as:
[tex]\( 81 = 3^4 \)[/tex]
2. Apply the cube root:
The given expression is [tex]\( \sqrt[3]{81} \)[/tex], which we now write as [tex]\( \sqrt[3]{3^4} \)[/tex].
3. Utilize the property of radicals:
The cube root of [tex]\( 3^4 \)[/tex] can be expressed using the property [tex]\( \sqrt[3]{a^b} = a^{\frac{b}{3}} \)[/tex]:
[tex]\[
\sqrt[3]{3^4} = 3^{\frac{4}{3}}
\][/tex]
Therefore, the simplified radical form of [tex]\( \sqrt[3]{81} \)[/tex] is [tex]\( 3^{\frac{4}{3}} \)[/tex].
Finally, evaluating [tex]\( 3^{\frac{4}{3}} \)[/tex] gives us the numeric value of approximately [tex]\( 4.3267 \)[/tex] when evaluated further, but the simplified radical form remains as [tex]\( 3^{\frac{4}{3}} \)[/tex] for exactness.
