To find the expression equivalent to [tex]\(\sqrt[3]{x} \cdot \sqrt[3]{x^4}\)[/tex], we need to manipulate the radicals and express them using exponents. Here’s the step-by-step solution:
1. Rewrite the Radicals with Exponents:
- The expression [tex]\(\sqrt[3]{x}\)[/tex] can be rewritten as [tex]\(x^{\frac{1}{3}}\)[/tex] since the cube root of [tex]\(x\)[/tex] is equivalent to [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{1}{3}\)[/tex].
- Similarly, the expression [tex]\(\sqrt[3]{x^4}\)[/tex] can be rewritten as [tex]\(x^{\frac{4}{3}}\)[/tex] since the cube root of [tex]\(x^4\)[/tex] is equivalent to [tex]\(x^4\)[/tex] raised to the power of [tex]\(\frac{1}{3}\)[/tex].
2. Combine the Exponents:
- When multiplying expressions with the same base, you add the exponents. Here, we are multiplying [tex]\(x^{\frac{1}{3}}\)[/tex] and [tex]\(x^{\frac{4}{3}}\)[/tex]:
[tex]\[
x^{\frac{1}{3}} \cdot x^{\frac{4}{3}} = x^{\frac{1}{3} + \frac{4}{3}}
\][/tex]
- Adding the exponents:
[tex]\[
\frac{1}{3} + \frac{4}{3} = \frac{1 + 4}{3} = \frac{5}{3}
\][/tex]
3. Equivalent Expression:
- So, the equivalent expression for [tex]\(\sqrt[3]{x} \cdot \sqrt[3]{x^4}\)[/tex] is [tex]\(x^{\frac{5}{3}}\)[/tex].
Given the choices:
(a) [tex]\(x^{\frac{5}{3}}\)[/tex]
(b) [tex]\(x^{\frac{3}{5}}\)[/tex]
(c) [tex]\(x^{\frac{4}{9}}\)[/tex]
The correct answer is:
[tex]\[
\boxed{a}
\][/tex]
Therefore, [tex]\(x^{\frac{5}{3}}\)[/tex] is the expression equivalent to [tex]\(\sqrt[3]{x} \cdot \sqrt[3]{x^4}\)[/tex] for all values of [tex]\(x\)[/tex].
