High School

The expression [tex]\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}[/tex] is equivalent to:

A. [tex]5^{\frac{25}{12}}[/tex]
B. [tex]5^{\frac{24}{25}}[/tex]
C. [tex]5^{\frac{12}{25}}[/tex]
D. [tex]5^{\frac{25}{24}}[/tex]

Answer :

To solve the expression [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex], we'll use the properties of exponents.

1. Rewrite the Radicals as Exponents:
- [tex]\(\sqrt[4]{5^5}\)[/tex] is the same as [tex]\(5^{5/4}\)[/tex].
- [tex]\(\sqrt[6]{5^5}\)[/tex] is the same as [tex]\(5^{5/6}\)[/tex].

2. Combine the Exponents:
- When multiplying expressions with the same base, you can add the exponents:
[tex]\[
5^{5/4} \cdot 5^{5/6} = 5^{(5/4) + (5/6)}
\][/tex]

3. Add the Exponents:
- To add the fractions [tex]\(\frac{5}{4}\)[/tex] and [tex]\(\frac{5}{6}\)[/tex], we need a common denominator. The least common denominator of 4 and 6 is 12.
- Convert each fraction:
- [tex]\(\frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12}\)[/tex]
- [tex]\(\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}\)[/tex]

- Now add the fractions:
[tex]\[
\frac{15}{12} + \frac{10}{12} = \frac{25}{12}
\][/tex]

4. Final Expression:
- When combined, the expression is equivalent to [tex]\(5^{\frac{25}{12}}\)[/tex].

So, the original expression [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex] simplifies to:

[tex]\[
5^{\frac{25}{12}}
\][/tex]

Thus, the correct answer is [tex]\(5^{\frac{25}{12}}\)[/tex].