Question : Simplify: $\frac{256 x^4-16 y^4}{\left(80 x^2-20 y^2\right)\left(16 x^2+4 y^2\right)}$
Option 1: $5$
Option 2: $\frac{1}{20}$
Option 3: $\frac{1}{5}$
Option 4: $\frac{2}{5}$
Correct Answer: $\frac{1}{5}$
Solution :
To simplify the given expression, let's factorize the numerator and the denominator first:
Numerator $=256x^4-16y^4$
$=16(16x^4-y^4)$
$=16(4x^2+y^2)(4x^2-y^2)$
Denominator $=\left(80x^2-20y^2\right)\left(16x^2+4 y^2\right)$
$=20(4x^2-y^2)×4(4x^2+y^2)$
$=80(4x^2+y^2)(4x^2–y^2)$
So, now $\frac{256x^4–16y^4}{\left(80x^2–20y^2\right)\left(16x^2+4 y^2\right)}$
$=\frac{16(4x^2+y^2)(4x^2-y^2)}{80(4x^2+y^2)(4x^2–y^2)}$
$=\frac{1}{5}$
Hence, the correct answer is $\frac{1}{5}$.
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