Sure! Let's simplify the given expression step-by-step:
We start with the expression:
[tex]\[
\left(\frac{y^4}{3 z^3}\right)^2
\][/tex]
### Step 1: Apply the exponent to the numerator and the denominator
When raising a fraction to a power, we raise both the numerator and the denominator to that power. So we get:
[tex]\[
\left(\frac{y^4}{3 z^3}\right)^2 = \frac{(y^4)^2}{(3 z^3)^2}
\][/tex]
### Step 2: Simplify each component in the fraction
Next, we simplify the numerator and the denominator separately.
#### Simplify the numerator:
[tex]\[
(y^4)^2 = y^{4 \cdot 2} = y^8
\][/tex]
#### Simplify the denominator:
[tex]\[
(3 z^3)^2 = 3^2 \cdot (z^3)^2 = 9 \cdot z^{3 \cdot 2} = 9 z^6
\][/tex]
### Step 3: Combine the simplified parts
Now, recombine the simplified numerator and denominator:
[tex]\[
\frac{y^8}{9 z^6}
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
\frac{y^8}{9 z^6}
\][/tex]
