Answer :
Sure! Let's break down the expression [tex]\(\sqrt[4]{81 x^8 y^6}\)[/tex] step-by-step to find its equivalent form.
1. Simplify the Constant Part:
- We start with the number 81. We know that 81 can be expressed as [tex]\(3^4\)[/tex]. Therefore, the fourth root of 81 is:
[tex]\[
\sqrt[4]{81} = \sqrt[4]{3^4} = 3
\][/tex]
2. Simplify the Variable [tex]\(x\)[/tex] Part:
- For [tex]\(x^8\)[/tex], we need to find [tex]\(\sqrt[4]{x^8}\)[/tex]. This can be rewritten as:
[tex]\[
(x^8)^{\frac{1}{4}} = x^{\frac{8}{4}} = x^2
\][/tex]
3. Simplify the Variable [tex]\(y\)[/tex] Part:
- For [tex]\(y^6\)[/tex], we find the fourth root as follows:
[tex]\[
(y^6)^{\frac{1}{4}} = y^{\frac{6}{4}} = y^{\frac{3}{2}}
\][/tex]
Putting it all together, the expression [tex]\(\sqrt[4]{81 x^8 y^6}\)[/tex] simplifies to:
[tex]\[
3 x^2 y^{\frac{3}{2}}
\][/tex]
Thus, the correct answer is [tex]\(3 x^2 y^{\frac{3}{2}}\)[/tex], and this matches none of the given options exactly as provided but aligns with the process.
1. Simplify the Constant Part:
- We start with the number 81. We know that 81 can be expressed as [tex]\(3^4\)[/tex]. Therefore, the fourth root of 81 is:
[tex]\[
\sqrt[4]{81} = \sqrt[4]{3^4} = 3
\][/tex]
2. Simplify the Variable [tex]\(x\)[/tex] Part:
- For [tex]\(x^8\)[/tex], we need to find [tex]\(\sqrt[4]{x^8}\)[/tex]. This can be rewritten as:
[tex]\[
(x^8)^{\frac{1}{4}} = x^{\frac{8}{4}} = x^2
\][/tex]
3. Simplify the Variable [tex]\(y\)[/tex] Part:
- For [tex]\(y^6\)[/tex], we find the fourth root as follows:
[tex]\[
(y^6)^{\frac{1}{4}} = y^{\frac{6}{4}} = y^{\frac{3}{2}}
\][/tex]
Putting it all together, the expression [tex]\(\sqrt[4]{81 x^8 y^6}\)[/tex] simplifies to:
[tex]\[
3 x^2 y^{\frac{3}{2}}
\][/tex]
Thus, the correct answer is [tex]\(3 x^2 y^{\frac{3}{2}}\)[/tex], and this matches none of the given options exactly as provided but aligns with the process.
