Of course! Let's simplify the given expression step by step using the exponential properties.
Given expression:
[tex]\[ \left(\frac{-3x^3y}{x^2}\right)^4 \][/tex]
### Step 1: Simplify inside the parentheses
First, look at the fraction inside the parentheses:
[tex]\[ \frac{-3x^3y}{x^2} \][/tex]
Using the property of exponents:
[tex]\[ \frac{x^m}{x^n} = x^{m-n} \][/tex]
we can simplify [tex]\( \frac{x^3}{x^2} \)[/tex] as follows:
[tex]\[ \frac{x^3}{x^2} = x^{3-2} = x \][/tex]
So, the expression inside the parentheses becomes:
[tex]\[ -3xy \][/tex]
### Step 2: Apply the power of a product property
Now, we have:
[tex]\[ (-3xy)^4 \][/tex]
Using the property:
[tex]\[ (ab)^n = a^n b^n \][/tex]
we can separate it as follows:
[tex]\[ (-3xy)^4 = (-3)^4 \cdot (x)^4 \cdot (y)^4 \][/tex]
### Step 3: Simplify constants and apply the power property
Next, we compute [tex]\( (-3)^4 \)[/tex]. Using:
[tex]\[ (-a)^n = a^n \text{ if n is even} \][/tex]
We get:
[tex]\[ (-3)^4 = 3^4 = 81 \][/tex]
So the expression becomes:
[tex]\[ (-3)^4 \cdot x^4 \cdot y^4 = 81 \cdot x^4 \cdot y^4 \][/tex]
This simplifies to:
[tex]\[ 81x^4y^4 \][/tex]
### Final Simplified Answer
The final simplified expression is:
[tex]\[ 81x^4y^4 \][/tex]
Thus, the given expression:
[tex]\[ \left(\frac{-3x^3y}{x^2}\right)^4 \][/tex]
simplifies to:
[tex]\[ 81x^4y^4 \][/tex]
Hence, the final simplified answer is:
[tex]\[ 81x^4y^4 \][/tex]
