Certainly! Let's simplify the expression step by step:
Given the expression:
[tex]$\left(\frac{a^3 b^{-4}}{2 a^1}\right)^{-2}$[/tex]
Step 1: Simplify the fraction inside the parentheses.
[tex]$\frac{a^3 b^{-4}}{2 a}$[/tex]
First, simplify the powers of [tex]\(a\)[/tex]:
[tex]$\frac{a^3}{a^1} = a^{3-1} = a^2$[/tex]
Now, substitute back into the expression:
[tex]$\frac{a^2 b^{-4}}{2}$[/tex]
Step 2: Re-write the expression:
[tex]$\left(\frac{a^2 b^{-4}}{2}\right)^{-2}$[/tex]
Step 3: Take the reciprocal of the fraction and change the sign of the exponent:
[tex]$\left(\frac{2}{a^2 b^{-4}}\right)^{2}$[/tex]
Step 4: Simplify the expression inside the parentheses again. Recall that [tex]\(b^{-4}\)[/tex] can be rewritten with a positive exponent:
[tex]$b^{-4} = \frac{1}{b^4}$[/tex]
Thus,
[tex]$\frac{2}{a^2 b^{-4}} = \frac{2}{a^2 \cdot \frac{1}{b^4}} = \frac{2 \cdot b^4}{a^2}$[/tex]
Step 5: Simplify the expression:
[tex]$\left(\frac{2 b^4}{a^2}\right)^{2}$[/tex]
Step 6: Apply the exponent to both the numerator and the denominator:
[tex]$\frac{(2 b^4)^2}{(a^2)^2}$[/tex]
Step 7: Simplify each part:
[tex]$(2 b^4)^2 = 2^2 \cdot (b^4)^2 = 4 b^8$[/tex]
[tex]$(a^2)^2 = a^{2 \cdot 2} = a^4$[/tex]
Step 8: Write the final simplified expression:
[tex]$\frac{4 b^8}{a^4}$[/tex]
Thus, the simplified expression with positive exponents is:
[tex]$\boxed{\frac{4 b^8}{a^4}}$[/tex]
