Which expressions are equivalent to [tex]\frac{10}{10^{\frac{3}{4}}}[/tex]?

A. [tex]10^{\frac{4}{3}}[/tex]
B. [tex]10^{\frac{1}{4}}[/tex]
C. [tex]\sqrt[3]{10^4}[/tex]
D. [tex]\sqrt[4]{10}[/tex]



Answer :

To determine which expressions are equivalent to [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex], let's analyze each option step-by-step.

First, let's simplify the given expression:

[tex]\[ \frac{10}{10^{\frac{3}{4}}} \][/tex]

We can rewrite the expression as:

[tex]\[ 10 \times 10^{-\frac{3}{4}} = 10^{1 - \frac{3}{4}} = 10^{\frac{4}{4} - \frac{3}{4}} = 10^{\frac{1}{4}} \][/tex]

Thus, [tex]\(\frac{10}{10^{\frac{3}{4}}} = 10^{\frac{1}{4}}\)[/tex].

Now, let's compare this with each of the given options:

1. [tex]\(10^{\frac{4}{3}}\)[/tex]

This is not equivalent to [tex]\(10^{\frac{1}{4}}\)[/tex].

2. [tex]\(10^{\frac{1}{4}}\)[/tex]

This is equivalent to [tex]\(10^{\frac{1}{4}}\)[/tex].

3. [tex]\(\sqrt[3]{10^4}\)[/tex]

We can rewrite [tex]\(\sqrt[3]{10^4}\)[/tex] as:

[tex]\[ (10^4)^{\frac{1}{3}} = 10^{\frac{4}{3}} \][/tex]

This is not equivalent to [tex]\(10^{\frac{1}{4}}\)[/tex].

4. [tex]\(\sqrt[4]{10}\)[/tex]

We can rewrite [tex]\(\sqrt[4]{10}\)[/tex] as:

[tex]\[ 10^{\frac{1}{4}} \][/tex]

This is equivalent to [tex]\(10^{\frac{1}{4}}\)[/tex].

Therefore, the equivalent expressions to [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex] are:
[tex]\[ 10^{\frac{1}{4}} \quad \text{and} \quad \sqrt[4]{10}. \][/tex]