Question

243^1/3 in simplest radical form

Answer 1

To simplify \( 243^{1/3} \) in simplest radical form, we first find the prime factorization of 243.

We can express 243 as:
\[
243 = 3^5
\]

Now, we can express \( 243^{1/3} \) using the prime factorization:
\[
243^{1/3} = (3^5)^{1/3}
\]

Using the exponent rule \((a^m)^n = a^{m \cdot n}\), we can simplify this:
\[
(3^5)^{1/3} = 3^{5/3}
\]

Now, we can split the exponent into whole and fractional parts:
\[
3^{5/3} = 3^{1 + 2/3} = 3^1 \cdot 3^{2/3} = 3 \cdot 3^{2/3}
\]

Finally, we can express \( 3^{2/3} \) in radical form:
\[
3^{2/3} = \sqrt[3]{3^2} = \sqrt[3]{9}
\]

Thus, we have:
\[
243^{1/3} = 3 \cdot \sqrt[3]{9}
\]

In conclusion, the simplest radical form of \( 243^{1/3} \) is:
\[
\boxed{3\sqrt[3]{9}}
\]