\(\displaystyle\frac{256\times256\times256-144\times144\times144}{256\times256+256\times144+144\times144}\) is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 112
Multiplying out these large numbers is not needed. The expression has the form of the identity \(a^3-b^3=(a-b)(a^2+ab+b^2)\), with \(a=256\) and \(b=144\), so the whole denominator cancels.
The numerator is a difference of cubes: \(256^3-144^3=a^3-b^3\)
The denominator is the second factor of the identity: \(256^2+256\times144+144^2=a^2+ab+b^2\)
So the expression reduces to \(\displaystyle\frac{(a-b)(a^2+ab+b^2)}{a^2+ab+b^2}=a-b\)
Its value is \(256-144=112\)
Tip: When the denominator is \(a^2+ab+b^2\) and the numerator is \(a^3-b^3\), the answer is \(a-b\) at once; with \(a^2-ab+b^2\) below \(a^3+b^3\), it is \(a+b\).