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 :

Let's examine which of the given expressions are equivalent to [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex].

First, simplify the expression [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex]:

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

Using the properties of exponents, we can combine the powers of 10:

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

So, the expression [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex] simplifies to [tex]\(10^{\frac{1}{4}}\)[/tex].

Now, let's compare this result with the given expressions:

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

Start by rewriting the given expressions in terms of exponents for consistency:

1. [tex]\(10^{\frac{4}{3}}\)[/tex] remains as it is.
2. [tex]\(10^{\frac{1}{4}}\)[/tex] is [tex]\(10^{\frac{1}{4}}\)[/tex], which matches our simplified expression [tex]\(10^{\frac{1}{4}}\)[/tex].
3. [tex]\(\sqrt[3]{10^4}\)[/tex] can be written as [tex]\((10^4)^{\frac{1}{3}} = 10^{\frac{4}{3}}\)[/tex].
4. [tex]\(\sqrt[4]{10}\)[/tex] can be written as [tex]\(10^{\frac{1}{4}}\)[/tex], which matches our simplified expression [tex]\(10^{\frac{1}{4}}\)[/tex].

From this, we see that expressions [tex]\(10^{\frac{1}{4}}\)[/tex] and [tex]\(\sqrt[4]{10}\)[/tex], which can be rewritten as [tex]\(10^{\frac{1}{4}}\)[/tex], match the original expression.

Therefore, the expressions equivalent to [tex]\(\frac{10}{10^{\frac{3}{4}}}\)[/tex] are:

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

In the given list, these correspond to:

- [tex]\(10^{\frac{1}{4}}\)[/tex] (the second expression)
- [tex]\(\sqrt[4]{10}\)[/tex] (the fourth expression)

Thus, the equivalent expressions are:

[tex]\[ 2, 4 \][/tex]