Answer :
Sure! Let's solve the expression [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex] step-by-step together:
1. Understanding the Expression:
The expression is given as [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex]. We need to simplify this expression.
2. Convert to Exponential Form:
Both terms involve roots which can be converted to fractional exponents:
[tex]\[
\sqrt[4]{5^5} = (5^5)^{\frac{1}{4}}
\][/tex]
[tex]\[
\sqrt[6]{5^5} = (5^5)^{\frac{1}{6}}
\][/tex]
3. Simplify Each Term:
Using the properties of exponents [tex]\((a^b)^c = a^{b \cdot c}\)[/tex], we can simplify each term:
[tex]\[
(5^5)^{\frac{1}{4}} = 5^{5 \cdot \frac{1}{4}} = 5^{\frac{5}{4}}
\][/tex]
[tex]\[
(5^5)^{\frac{1}{6}} = 5^{5 \cdot \frac{1}{6}} = 5^{\frac{5}{6}}
\][/tex]
4. Combine the Terms:
Now that we have:
[tex]\[
5^{\frac{5}{4}} \cdot 5^{\frac{5}{6}}
\][/tex]
We use the property of exponents that says [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
5^{\frac{5}{4}} \cdot 5^{\frac{5}{6}} = 5^{\frac{5}{4} + \frac{5}{6}}
\][/tex]
5. Add the Exponents:
To add the exponents, we need a common denominator. The denominators are 4 and 6. The least common multiple of 4 and 6 is 12.
[tex]\[
\frac{5}{4} = \frac{5 \cdot 3}{4 \cdot 3} = \frac{15}{12}
\][/tex]
[tex]\[
\frac{5}{6} = \frac{5 \cdot 2}{6 \cdot 2} = \frac{10}{12}
\][/tex]
Now, add them together:
[tex]\[
\frac{15}{12} + \frac{10}{12} = \frac{15 + 10}{12} = \frac{25}{12}
\][/tex]
6. Final Expression:
Combining it all together, we get:
[tex]\[
5^{\frac{25}{12}}
\][/tex]
Therefore, the given expression [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex] is equivalent to [tex]\(5^{\frac{25}{12}}\)[/tex]. The correct answer is:
[tex]\[
\boxed{5^{\frac{25}{12}}}
\][/tex]
1. Understanding the Expression:
The expression is given as [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex]. We need to simplify this expression.
2. Convert to Exponential Form:
Both terms involve roots which can be converted to fractional exponents:
[tex]\[
\sqrt[4]{5^5} = (5^5)^{\frac{1}{4}}
\][/tex]
[tex]\[
\sqrt[6]{5^5} = (5^5)^{\frac{1}{6}}
\][/tex]
3. Simplify Each Term:
Using the properties of exponents [tex]\((a^b)^c = a^{b \cdot c}\)[/tex], we can simplify each term:
[tex]\[
(5^5)^{\frac{1}{4}} = 5^{5 \cdot \frac{1}{4}} = 5^{\frac{5}{4}}
\][/tex]
[tex]\[
(5^5)^{\frac{1}{6}} = 5^{5 \cdot \frac{1}{6}} = 5^{\frac{5}{6}}
\][/tex]
4. Combine the Terms:
Now that we have:
[tex]\[
5^{\frac{5}{4}} \cdot 5^{\frac{5}{6}}
\][/tex]
We use the property of exponents that says [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
5^{\frac{5}{4}} \cdot 5^{\frac{5}{6}} = 5^{\frac{5}{4} + \frac{5}{6}}
\][/tex]
5. Add the Exponents:
To add the exponents, we need a common denominator. The denominators are 4 and 6. The least common multiple of 4 and 6 is 12.
[tex]\[
\frac{5}{4} = \frac{5 \cdot 3}{4 \cdot 3} = \frac{15}{12}
\][/tex]
[tex]\[
\frac{5}{6} = \frac{5 \cdot 2}{6 \cdot 2} = \frac{10}{12}
\][/tex]
Now, add them together:
[tex]\[
\frac{15}{12} + \frac{10}{12} = \frac{15 + 10}{12} = \frac{25}{12}
\][/tex]
6. Final Expression:
Combining it all together, we get:
[tex]\[
5^{\frac{25}{12}}
\][/tex]
Therefore, the given expression [tex]\(\sqrt[4]{5^5} \cdot \sqrt[6]{5^5}\)[/tex] is equivalent to [tex]\(5^{\frac{25}{12}}\)[/tex]. The correct answer is:
[tex]\[
\boxed{5^{\frac{25}{12}}}
\][/tex]