Let's simplify the expression [tex]\(\left(729 x^3 y^{20} z^{11}\right)^{\frac{1}{5}}\)[/tex] step-by-step.
1. Understand the given expression:
[tex]\[
\left(729 x^3 y^{20} z^{11}\right)^{\frac{1}{5}}
\][/tex]
2. Break down the expression inside the parenthesis:
[tex]\[
729 = 3^6
\][/tex]
So the expression becomes:
[tex]\[
(3^6 x^3 y^{20} z^{11})^{\frac{1}{5}}
\][/tex]
3. Apply the exponent [tex]\(\frac{1}{5}\)[/tex] to each term inside the parenthesis individually:
According to the laws of exponents:
[tex]\[
(a \cdot b \cdot c)^{n} = a^n \cdot b^n \cdot c^n
\][/tex]
Therefore:
[tex]\[
(3^6)^{\frac{1}{5}} \cdot (x^3)^{\frac{1}{5}} \cdot (y^{20})^{\frac{1}{5}} \cdot (z^{11})^{\frac{1}{5}}
\][/tex]
4. Simplify each term:
[tex]\[
(3^6)^{\frac{1}{5}} = 3^{\frac{6}{5}}
\][/tex]
[tex]\[
(x^3)^{\frac{1}{5}} = x^{3 \cdot \frac{1}{5}} = x^{\frac{3}{5}}
\][/tex]
[tex]\[
(y^{20})^{\frac{1}{5}} = y^{20 \cdot \frac{1}{5}} = y^4
\][/tex]
[tex]\[
(z^{11})^{\frac{1}{5}} = z^{11 \cdot \frac{1}{5}} = z^{\frac{11}{5}}
\][/tex]
5. Combine all the simplified terms:
[tex]\[
3^{\frac{6}{5}} \cdot x^{\frac{3}{5}} \cdot y^4 \cdot z^{\frac{11}{5}}
\][/tex]
6. Express in a form involving the 5th root where necessary:
We can rewrite [tex]\(3^{\frac{6}{5}}\)[/tex] as [tex]\(3 \cdot 3^{\frac{1}{5}}\)[/tex]:
[tex]\[
3 \cdot 3^{\frac{1}{5}} \cdot y^4 \cdot z^2 \cdot (xz)^{\frac{1}{5}}
\][/tex]
Here, [tex]\(z^{\frac{11}{5}}\)[/tex] can be written as [tex]\(z^2 \cdot z^{\frac{1}{5}}\)[/tex]
7. Identify the correct matching option:
Looking at the provided options:
A. [tex]\(3 y^4 z^2 \sqrt[5]{3 x^3 z}\)[/tex]
B. [tex]\(27 y^{15} z^6 \sqrt[5]{3 x^3}\)[/tex]
C. [tex]\(27 y^4 z^2 \sqrt[5]{x^3 z}\)[/tex]
D. [tex]\(3 y^{15} z^6 \sqrt[5]{3 x^3}\)[/tex]
The correct answer is:
A. [tex]\(3 y^4 z^2 \sqrt[5]{3 x^3 z}\)[/tex]
