Certainly! Let's simplify the given expression step by step:
Given expression:
[tex]\[
\left(\frac{b^4}{-4 a^2}\right)^3
\][/tex]
### Step 1: Simplify inside the parentheses
First, simplify the fraction inside the parentheses:
[tex]\[
\frac{b^4}{-4 a^2} = \frac{-b^4}{4 a^2}
\][/tex]
### Step 2: Raise the simplified fraction to the power of 3
Now, raise the simplified fraction to the power of 3:
[tex]\[
\left(\frac{-b^4}{4 a^2}\right)^3
\][/tex]
### Step 3: Distribute the power to both the numerator and the denominator
When you raise a fraction to a power, you raise both the numerator and the denominator to that power:
[tex]\[
\left(\frac{-b^4}{4 a^2}\right)^3 = \left( -b^4 \right)^3 \Bigg/ \left( 4 a^2 \right)^3
\][/tex]
### Step 4: Simplify the numerator and the denominator separately
Simplify the numerator:
[tex]\[
\left( -b^4 \right)^3 = (-1)^3 \cdot (b^4)^3 = - (b^{4 \cdot 3}) = - b^{12}
\][/tex]
Simplify the denominator:
[tex]\[
(4 a^2)^3 = 4^3 \cdot (a^2)^3 = 64 \cdot a^{2 \cdot 3} = 64 a^6
\][/tex]
### Step 5: Combine the simplified numerator and denominator
Now, combining the simplified numerator and denominator, we get:
[tex]\[
\frac{-b^{12}}{64 a^6}
\][/tex]
So, the simplified expression is:
[tex]\[
-\frac{b^{12}}{64 a^6}
\][/tex]
In a more compact form without the parentheses, it is written as:
[tex]\[
-b^{12} / 64a^6
\][/tex]
Therefore, the simplified expression is:
[tex]\[
-\frac{b^{12}}{64a^6}
\][/tex]
