Question

which expression is equivalent to $x^{\frac{3}{2}}\cdot x^{\frac{1}{4}}$?\
\bigcirc $\sqrt{x^3}\cdot\sqrt4{x}$\
\bigcirc $\sqrt7{x^2}$\
\bigcirc $\sqrt{x^7}$\
\bigcirc $x^{\frac{7}{2}}$

Explanation:

Step1: Use exponent rule for multiplication

When multiplying exponents with the same base, we add the exponents. So for \( x^{\frac{3}{2}} \cdot x^{\frac{1}{4}} \), we add \( \frac{3}{2} \) and \( \frac{1}{4} \).
\( \frac{3}{2}+\frac{1}{4}=\frac{6}{4}+\frac{1}{4}=\frac{7}{4} \), so the expression simplifies to \( x^{\frac{7}{4}} \).

Step2: Analyze each option

• Option 1: \( \sqrt{x^3} \cdot \sqrt[4]{x} \). Rewrite radicals as exponents: \( \sqrt{x^3}=x^{\frac{3}{2}} \) and \( \sqrt[4]{x}=x^{\frac{1}{4}} \). Multiply them: \( x^{\frac{3}{2}} \cdot x^{\frac{1}{4}} = x^{\frac{3}{2}+\frac{1}{4}} = x^{\frac{7}{4}} \), which matches our result.
• Option 2: \( \sqrt[7]{x^2}=x^{\frac{2}{7}} \), not equal to \( x^{\frac{7}{4}} \).
• Option 3: \( \sqrt{x^7}=x^{\frac{7}{2}} \), not equal to \( x^{\frac{7}{4}} \).
• Option 4: \( x^{\frac{7}{2}} \), not equal to \( x^{\frac{7}{4}} \).

Answer:

A. \(\sqrt{x^3} \cdot \sqrt[4]{x}\)