To simplify the expression [tex]\(\frac{x^3 y^{\frac{1}{2}} z^5}{x^{-\frac{4}{3}}\left(y^2 z\right)^5}\)[/tex], we need to break it down step-by-step and use the laws of exponents.
Given expression:
[tex]\[
\frac{x^3 y^{\frac{1}{2}} z^5}{x^{-\frac{4}{3}}\left(y^2 z\right)^5}
\][/tex]
First, let's simplify the denominator:
[tex]\[
x^{-\frac{4}{3}} \left(y^2 z\right)^5
\][/tex]
Using the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can expand [tex]\(\left(y^2 z\right)^5\)[/tex]:
[tex]\[
\left(y^2 z\right)^5 = (y^2)^5 \cdot z^5 = y^{2 \cdot 5} \cdot z^5 = y^{10} \cdot z^5
\][/tex]
So now the denominator is:
[tex]\[
x^{-\frac{4}{3}} y^{10} z^5
\][/tex]
Now the expression becomes:
[tex]\[
\frac{x^3 y^{\frac{1}{2}} z^5}{x^{-\frac{4}{3}} y^{10} z^5}
\][/tex]
Next, we can separate the variables:
[tex]\[
\frac{x^3}{x^{-\frac{4}{3}}} \cdot \frac{y^{\frac{1}{2}}}{y^{10}} \cdot \frac{z^5}{z^5}
\][/tex]
Simplifying each fraction individually:
1. Simplify the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{x^3}{x^{-\frac{4}{3}}} = x^{3 - \left(-\frac{4}{3}\right)} = x^{3 + \frac{4}{3}} = x^{\frac{9}{3} + \frac{4}{3}} = x^{\frac{13}{3}}
\][/tex]
2. Simplify the [tex]\(y\)[/tex] terms:
[tex]\[
\frac{y^{\frac{1}{2}}}{y^{10}} = y^{\frac{1}{2} - 10} = y^{\frac{1}{2} - \frac{20}{2}} = y^{-\frac{19}{2}}
\][/tex]
3. Simplify the [tex]\(z\)[/tex] terms:
[tex]\[
\frac{z^5}{z^5} = z^{5 - 5} = z^0 = 1
\][/tex]
So combining all the simplified parts together, we get:
[tex]\[
x^{\frac{13}{3}} \cdot y^{-\frac{19}{2}} \cdot 1 = x^{\frac{13}{3}} y^{-\frac{19}{2}}
\][/tex]
Therefore, the expression equivalent to the given expression is:
[tex]\[
x^{\frac{13}{3}} y^{-\frac{19}{2}}
\][/tex]
The correct answer is:
A) [tex]\(x^{\frac{13}{3}} y^{-\frac{19}{2}}\)[/tex]
