Rewrite in simplest rational exponent form square root of x times the fourth root of x. Show each step of your process.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To rewrite the expression \(\sqrt{x} \times \sqrt[4]{x}\) in simplest rational exponent form, follow these steps:
1. Convert to Exponential Form:
The square root of \(x\) can be expressed as \(x^{1/2}\).
The fourth root of \(x\) can be expressed as \(x^{1/4}\).
So, we have:
\[
\sqrt{x} = x^{1/2} \quad \text{and} \quad \sqrt[4]{x} = x^{1/4}
\]
2. Multiply the Exponents:
Now, multiply the two expressions:
\[
x^{1/2} \times x^{1/4}
\]
3. Apply the Product of Powers Rule:
When multiplying like bases, add the exponents:
\[
x^{1/2 + 1/4}
\]
4. Find a Common Denominator:
The common denominator for \(1/2\) and \(1/4\) is \(4\):
\[
1/2 = 2/4 \quad \text{and} \quad 1/4 = 1/4
\]
5. Add the Exponents:
\[
2/4 + 1/4 = 3/4
\]
6. Final Expression:
Therefore, the expression simplifies to:
\[
x^{3/4}
\]
So, the simplest rational exponent form of \(\sqrt{x} \times \sqrt[4]{x}\) is \(x^{3/4}\).