Question

Rewrite in simplest rational exponent form square root of x times the fourth root of x. Show each step of your process.

Answer 1

To rewrite the expression \(\sqrt{x} \times \sqrt[4]{x}\) in simplest rational exponent form, follow these steps:

1. **Rewrite the roots using rational exponents**:
- The square root of \(x\) can be rewritten as \(x^{1/2}\).
- The fourth root of \(x\) can be rewritten as \(x^{1/4}\).

So we have:
\[
\sqrt{x} = x^{1/2}
\]
\[
\sqrt[4]{x} = x^{1/4}
\]

2. **Multiply the expressions**:
Now, substitute these forms back into the original expression:
\[
\sqrt{x} \times \sqrt[4]{x} = x^{1/2} \times x^{1/4}
\]

3. **Combine the exponents**:
When multiplying expressions with the same base, we add the exponents:
\[
x^{1/2} \times x^{1/4} = x^{1/2 + 1/4}
\]

4. **Find a common denominator**:
To add \(1/2\) and \(1/4\), we need a common denominator. The least common denominator of 2 and 4 is 4:
\[
\frac{1}{2} = \frac{2}{4}
\]

Now we can add:
\[
\frac{2}{4} + \frac{1}{4} = \frac{3}{4}
\]

5. **Write the final expression**:
Substitute the sum back into the exponent:
\[
x^{1/2 + 1/4} = x^{3/4}
\]

Thus, the expression \(\sqrt{x} \times \sqrt[4]{x}\) in simplest rational exponent form is \(\boxed{x^{3/4}}\).