Question

which expressions are equivalent to \\(\frac{10}{10^{\frac{3}{4}}}\\)? choose two correct answers. \\(\sqrt3{10^4}\\) \\(10^{\frac{4}{3}}\\) \\(\sqrt4{10}\\) \\(10^{\frac{1}{4}}\\)

Explanation:

Step1: Simplify the given expression

Using the exponent rule \( \frac{a^m}{a^n}=a^{m - n} \), for \( \frac{10}{10^{\frac{3}{4}}} \), we have \( 10^{1-\frac{3}{4}}=10^{\frac{1}{4}} \).

Step2: Analyze each option

• For \( \sqrt[3]{10^4} \): This is \( 10^{\frac{4}{3}} \), not equivalent.
• For \( 10^{\frac{4}{3}} \): Not equivalent to \( 10^{\frac{1}{4}} \).
• For \( \sqrt[4]{10} \): By the definition of radicals, \( \sqrt[n]{a}=a^{\frac{1}{n}} \), so \( \sqrt[4]{10}=10^{\frac{1}{4}} \), equivalent.
• For \( 10^{\frac{1}{4}} \): Matches the simplified form, equivalent.

Answer:

\( \sqrt[4]{10} \), \( 10^{\frac{1}{4}} \)