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 \(\frac{10}{10^{\frac{3}{4}}}\), we will simplify this expression step-by-step and then compare it to the given ones.

1. Simplify \(\frac{10}{10^{\frac{3}{4}}}\):

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

So, the simplified form of \(\frac{10}{10^{\frac{3}{4}}}\) is \(10^{\frac{1}{4}}\).

2. Compare the simplified form \(10^{\frac{1}{4}}\) to the given expressions:

- \(10^{\frac{4}{3}}\):
[tex]\[ 10^{\frac{4}{3}} \neq 10^{\frac{1}{4}} \][/tex]
This expression is not equivalent to our simplified form.

- \(10^{\frac{1}{4}}\):
[tex]\[ 10^{\frac{1}{4}} = 10^{\frac{1}{4}} \][/tex]
This expression is equivalent to our simplified form.

- \(\sqrt[3]{10^4}\):
[tex]\[ \sqrt[3]{10^4} = (10^4)^{\frac{1}{3}} = 10^{\frac{4}{3}} \neq 10^{\frac{1}{4}} \][/tex]
This expression is not equivalent to our simplified form.

- \(\sqrt[4]{10}\):
[tex]\[ \sqrt[4]{10} = 10^{\frac{1}{4}} = 10^{\frac{1}{4}} \][/tex]
This expression is equivalent to our simplified form.

Based on this comparison, the expressions equivalent to \(\frac{10}{10^{\frac{3}{4}}}\) are:

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

Thus, the correct options are:
- \(10^{\frac{1}{4}}\)
- [tex]\(\sqrt[4]{10}\)[/tex]