Certainly! Let's rewrite the expression [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] in its simplest rational exponent form step by step.
1. Rewrite the radicals as exponents:
- The square root of [tex]\(x\)[/tex] can be written as [tex]\(x^{1/2}\)[/tex].
- The fourth root of [tex]\(x\)[/tex] can be written as [tex]\(x^{1/4}\)[/tex].
So, the expression [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] can be rewritten as:
[tex]\[
x^{1/2} \cdot x^{1/4}
\][/tex]
2. Apply the product rule for exponents:
- When you multiply expressions with the same base, you can add the exponents.
Given the expression [tex]\(x^{1/2} \cdot x^{1/4}\)[/tex]:
[tex]\[
x^{1/2 + 1/4}
\][/tex]
3. Add the exponents:
- To add the exponents [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex], you need a common denominator.
- The common denominator for [tex]\(2\)[/tex] and [tex]\(4\)[/tex] is [tex]\(4\)[/tex], so we convert [tex]\(\frac{1}{2}\)[/tex] to [tex]\(\frac{2}{4}\)[/tex].
[tex]\[
\frac{1}{2} = \frac{2}{4}
\][/tex]
- Now, add [tex]\(\frac{2}{4}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\frac{2}{4} + \frac{1}{4} = \frac{3}{4}
\][/tex]
4. Write the simplified expression with the new exponent:
[tex]\[
x^{3/4}
\][/tex]
Therefore, the simplest rational exponent form of [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] is:
[tex]\[
x^{3/4}
\][/tex]
