Certainly! Let's simplify the given expression step by step:
We start with the given expression:
[tex]\[ \frac{2 + \frac{b^4 - 4a^2c^2}{a^4} - \frac{c}{c^2}a^2}{\frac{c^2}{a^2}} \][/tex]
### Step 1: Simplify the Numerator
First, we need to simplify the numerator:
[tex]\[ 2 + \frac{b^4 - 4a^2c^2}{a^4} - \frac{c}{c^2}a^2 \][/tex]
Notice that [tex]\(\frac{c}{c^2}\)[/tex] simplifies to [tex]\(\frac{1}{c}\)[/tex], since [tex]\(c = c^1\)[/tex]:
[tex]\[ \frac{c}{c^2} = \frac{c}{c \cdot c} = \frac{1}{c} \][/tex]
Thus, the numerator becomes:
[tex]\[ 2 + \frac{b^4 - 4a^2c^2}{a^4} - \frac{a^2}{c} \][/tex]
### Step 2: Combine the Terms in the Numerator
We rewrite the numerator with a common denominator of [tex]\(a^4\)[/tex]:
[tex]\[ 2 + \frac{b^4 - 4a^2c^2}{a^4} - \frac{a^2 c}{c^2} \][/tex]
Simplify [tex]\(\frac{a^2 c}{c^2}\)[/tex] to [tex]\( \frac{a^2}{c} \)[/tex]:
[tex]\[ \frac{a^2}{c} = \frac{a^2}{c^2} \cdot c = a^2 \cdot \frac{1}{c} = a^2 \cdot \frac{1}{c^2} \cdot c = \frac{a^2}{1}= a^2\][/tex]
[tex]\[2 + \frac{b^4 - 4a^2c^2}{a^4} - a^2 \frac{1}{c}\][/tex]
Combine fractions:
[tex]\[2 + \frac{b^4 - 4a^2c^2 }{a^4} - a^2c \frac{a^2}{c^2}\][/tex]
### Step 3: Simplify the Denominator
The denominator is:
[tex]\[ \frac{c^2}{a^2} \][/tex]
### Step 4: Combine the Expression
Now, we have:
[tex]\[ \frac{2 + \frac{b^4 - 4a^2c^2}{a^4} - \frac{a^2}{c}}{\frac{c^2}{a^2}} \][/tex]
To simplify this, we multiply the numerator by the reciprocal of the denominator:
[tex]\[ \left(2 + \frac{b^4 - 4a^2c^2}{a^4} - \frac{a^2}{c} \right) \div \frac{c^2}{a^2} = \left(2 + \frac{b^4 - 4a^2c^2}{a^4} - \frac{a^2}{c} \right) \times \frac{a^2}{c^2} \][/tex]
Simplify this:
[tex]\[ \left(2 + \frac{b^4 - 4a^2c^2}{a^4} - \frac{a^2}{c} \right) \times \left(\frac{a^2}{c^2}\right) \][/tex]
Distribute [tex]\(\frac{a^2}{c^2}\)[/tex] to each term inside the parentheses:
[tex]\[ \left(2 \cdot \frac{a^2}{c^2} + \frac{b^4 - 4a^2c^2}{a^4} \cdot \frac{a^2}{c^2} - \frac{a^2}{c} \cdot \frac{a^2}{c^2} \right) \][/tex]
[tex]\[ = a^2 \left(2 \cdot \frac{1}{c^2} + \frac{b^4 - 4a^2c^2}{a^4} \cdot 1 - \frac{a^2}{c^3}\right) \][/tex]
Simplifying further, we have:
[tex]\[ a^2 \left(\frac{2}{c^2} + \frac{b^4 - 4a^2c^2}{a^4} - \frac{a^2}{c^3} \right) \][/tex]
Therefore, the simplified expression is:
[tex]\[ \boxed{a^2 \left( \frac{2}{c^2} + \frac{(b^4 - 4a^2c^2)/a^4} - \frac{a^2}{c^3}\right} )\][/tex]
