To rewrite the expression [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] in its simplest rational exponent form, we will follow these steps:
1. Express each root as a fractional exponent:
- The square root of [tex]\(x\)[/tex], [tex]\(\sqrt{x}\)[/tex], can be written as [tex]\(x^{1/2}\)[/tex].
- The fourth root of [tex]\(x\)[/tex], [tex]\(\sqrt[4]{x}\)[/tex], can be written as [tex]\(x^{1/4}\)[/tex].
Therefore, our original expression can be rewritten as:
[tex]\[
\sqrt{x} \cdot \sqrt[4]{x} = x^{1/2} \cdot x^{1/4}
\][/tex]
2. Apply the properties of exponents:
When multiplying expressions with the same base, we add the exponents. So, we need to add the exponents [tex]\(1/2\)[/tex] and [tex]\(1/4\)[/tex]:
[tex]\[
x^{1/2} \cdot x^{1/4} = x^{(1/2 + 1/4)}
\][/tex]
3. Find a common denominator to add the fractions:
- The fractions [tex]\(1/2\)[/tex] and [tex]\(1/4\)[/tex] need a common denominator to be added together.
- The common denominator for [tex]\(1/2\)[/tex] and [tex]\(1/4\)[/tex] is 4.
Convert [tex]\(1/2\)[/tex] to a fraction with denominator 4:
[tex]\[
1/2 = 2/4
\][/tex]
Now, add the fractions:
[tex]\[
2/4 + 1/4 = 3/4
\][/tex]
4. Combine the exponents:
Thus, the expression [tex]\(x^{1/2} \cdot x^{1/4}\)[/tex] simplifies to:
[tex]\[
x^{3/4}
\][/tex]
Therefore, the simplest rational exponent form of [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] is [tex]\(\boxed{x^{3/4}}\)[/tex].
