Question

Rewrite the radical form of 3√8x⁹ to the exponent form using rational exponents. Do not simplify.

Answer 1

To rewrite the expression \( \sqrt[3]{8x^9} \) in exponent form using rational exponents, we can express both the number and the variable with their respective exponents.

The cube root can be represented as raising to the power of \( \frac{1}{3} \). Thus, we have:

\[
\sqrt[3]{8x^9} = (8x^9)^{\frac{1}{3}}
\]

Now, we can express \( 8 \) as \( 2^3 \):

\[
(8x^9)^{\frac{1}{3}} = (2^3 x^9)^{\frac{1}{3}}
\]

Using the property of exponents \( (a^m b^n)^p = a^{mp} b^{np} \), we can distribute the exponent \( \frac{1}{3} \):

\[
(2^3)^{\frac{1}{3}} (x^9)^{\frac{1}{3}} = 2^{3 \cdot \frac{1}{3}} x^{9 \cdot \frac{1}{3}} = 2^1 x^{3}
\]

However, since you requested to not simplify the expression, we can leave it as:

\[
(8x^9)^{\frac{1}{3}}
\]

Thus, the final expression in exponent form using rational exponents is:

\[
8^{\frac{1}{3}} x^{\frac{9}{3}}
\]

Or simply, without simplification:

\[
(8x^9)^{\frac{1}{3}}
\]