Answer :
Let's solve the expression [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex].
1. Understand the Expressions with Roots and Exponents:
- [tex]\(\sqrt[4]{5^5}\)[/tex] can be rewritten using exponents as [tex]\((5^5)^{1/4}\)[/tex].
- Similarly, [tex]\(\sqrt[6]{5^5}\)[/tex] can be expressed as [tex]\((5^5)^{1/6}\)[/tex].
2. Simplify Each Term Separately:
- For [tex]\((5^5)^{1/4}\)[/tex], use the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(5^5)^{1/4} = 5^{5 \cdot \frac{1}{4}} = 5^{1.25}
\][/tex]
- For [tex]\((5^5)^{1/6}\)[/tex], similarly apply the power of a power property:
[tex]\[
(5^5)^{1/6} = 5^{5 \cdot \frac{1}{6}} = 5^{0.8333333333333333}
\][/tex]
3. Combine the Powers Using the Multiplication Rule:
- When multiplying expressions with the same base, you add the exponents: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
- Therefore, combine the exponents:
[tex]\[
5^{1.25} \cdot 5^{0.8333333333333333} = 5^{1.25 + 0.8333333333333333}
\][/tex]
- Add the exponents:
[tex]\[
1.25 + 0.8333333333333333 = 2.083333333333333
\][/tex]
4. Final Expression:
- The expression simplifies to:
[tex]\[
5^{2.083333333333333}
\][/tex]
Recognizing the result is a close variant of the fractional form [tex]\(5^{\frac{25}{12}}\)[/tex], which matches one of the options given in the problem. Therefore, [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex] is equivalent to [tex]\(5^{\frac{25}{12}}\)[/tex].
1. Understand the Expressions with Roots and Exponents:
- [tex]\(\sqrt[4]{5^5}\)[/tex] can be rewritten using exponents as [tex]\((5^5)^{1/4}\)[/tex].
- Similarly, [tex]\(\sqrt[6]{5^5}\)[/tex] can be expressed as [tex]\((5^5)^{1/6}\)[/tex].
2. Simplify Each Term Separately:
- For [tex]\((5^5)^{1/4}\)[/tex], use the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(5^5)^{1/4} = 5^{5 \cdot \frac{1}{4}} = 5^{1.25}
\][/tex]
- For [tex]\((5^5)^{1/6}\)[/tex], similarly apply the power of a power property:
[tex]\[
(5^5)^{1/6} = 5^{5 \cdot \frac{1}{6}} = 5^{0.8333333333333333}
\][/tex]
3. Combine the Powers Using the Multiplication Rule:
- When multiplying expressions with the same base, you add the exponents: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
- Therefore, combine the exponents:
[tex]\[
5^{1.25} \cdot 5^{0.8333333333333333} = 5^{1.25 + 0.8333333333333333}
\][/tex]
- Add the exponents:
[tex]\[
1.25 + 0.8333333333333333 = 2.083333333333333
\][/tex]
4. Final Expression:
- The expression simplifies to:
[tex]\[
5^{2.083333333333333}
\][/tex]
Recognizing the result is a close variant of the fractional form [tex]\(5^{\frac{25}{12}}\)[/tex], which matches one of the options given in the problem. Therefore, [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex] is equivalent to [tex]\(5^{\frac{25}{12}}\)[/tex].