Let's simplify the given expression step by step.
The given expression is:
[tex]\[
\frac{\left(2 x^2 - 5 y\right)^4}{\left(3 x^5\right)^3}
\][/tex]
First, let's rewrite both the numerator and the denominator to understand each part separately.
### Step 1: Rewrite the numerator
The numerator is [tex]\(\left(2 x^2 - 5 y\right)^4\)[/tex]. This expression is already in its simplified form, so there's no need to make any changes here.
### Step 2: Rewrite the denominator
The denominator is [tex]\(\left(3 x^5\right)^3\)[/tex]. We can simplify this expression by applying the exponent rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(3 x^5)^3 = 3^3 \cdot (x^5)^3
\][/tex]
Calculating the powers, we get:
[tex]\[
3^3 = 27 \quad \text{and} \quad (x^5)^3 = x^{5 \cdot 3} = x^{15}
\][/tex]
Thus, the denominator simplifies to:
[tex]\[
27 x^{15}
\][/tex]
### Step 3: Combine the simplified numerator and denominator
Now, we can write the simplified fraction:
[tex]\[
\frac{(2 x^2 - 5 y)^4}{27 x^{15}}
\][/tex]
Putting it all together, the simplified form of the given expression is:
[tex]\[
\frac{(2 x^2 - 5 y)^4}{27 x^{15}}
\][/tex]
This is the simplified version of the expression.
