Sure, let's rewrite \(\sqrt{x} \cdot \sqrt[4]{x}\) in the simplest rational exponent form step by step.
### Step 1: Express Each Root in Exponential Form
First, we need to express the square root and the fourth root in terms of exponents.
- The square root of \(x\) (\(\sqrt{x}\)) can be written as \(x^{1/2}\).
- The fourth root of \(x\) (\(\sqrt[4]{x}\)) can be written as \(x^{1/4}\).
Thus, the expression \(\sqrt{x} \cdot \sqrt[4]{x}\) can be rewritten as:
[tex]\[
x^{1/2} \cdot x^{1/4}
\][/tex]
### Step 2: Use Properties of Exponents to Combine
Next, we use the properties of exponents to combine these terms. Specifically, if we are multiplying two expressions with the same base, we add their exponents:
[tex]\[
x^{1/2} \cdot x^{1/4} = x^{1/2 + 1/4}
\][/tex]
### Step 3: Add the Exponents
Now, we need to add the fractions \( \frac{1}{2} \) and \( \frac{1}{4} \).
[tex]\[
\frac{1}{2} + \frac{1}{4}
\][/tex]
To add these fractions, we need a common denominator. The common denominator for 2 and 4 is 4. Therefore:
[tex]\[
\frac{1}{2} = \frac{2}{4}
\][/tex]
Now we can add:
[tex]\[
\frac{2}{4} + \frac{1}{4} = \frac{3}{4}
\][/tex]
### Step 4: Rewrite the Expression
Using the sum of the exponents, we can rewrite the original expression with a single exponent:
[tex]\[
x^{1/2 + 1/4} = x^{3/4}
\][/tex]
Thus, the simplest rational exponent form of \(\sqrt{x} \cdot \sqrt[4]{x}\) is:
[tex]\[
x^{3/4}
\][/tex]
The process shows that [tex]\(\sqrt{x} \cdot \sqrt[4]{x} = x^{3/4}\)[/tex].
