Sure! Let's simplify the expression [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] into its simplest rational exponent form step by step.
1. First, we rewrite [tex]\(\sqrt{x}\)[/tex] and [tex]\(\sqrt[4]{x}\)[/tex] using rational exponents.
- The square root of [tex]\(x\)[/tex] can be written as [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\sqrt{x} = x^{\frac{1}{2}}
\][/tex]
- The fourth root of [tex]\(x\)[/tex] can be written as [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
\sqrt[4]{x} = x^{\frac{1}{4}}
\][/tex]
2. Next, we multiply these two terms together:
[tex]\[
x^{\frac{1}{2}} \cdot x^{\frac{1}{4}}
\][/tex]
3. To multiply exponential terms with the same base, we add their exponents:
[tex]\[
x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} = x^{\left(\frac{1}{2} + \frac{1}{4}\right)}
\][/tex]
4. We now need to add the exponents [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
- First, find a common denominator. The least common denominator of 2 and 4 is 4:
[tex]\[
\frac{1}{2} = \frac{2}{4}
\][/tex]
- Now, add the fractions:
[tex]\[
\frac{2}{4} + \frac{1}{4} = \frac{3}{4}
\][/tex]
5. Therefore, [tex]\(\frac{1}{2} + \frac{1}{4} = \frac{3}{4}\)[/tex]. We can now write the expression with the simplified exponent:
[tex]\[
x^{\left(\frac{3}{4}\right)}
\][/tex]
Hence, the simplest rational exponent form of [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] is:
[tex]\[
x^{\frac{3}{4}}
\][/tex]
