If α and β be two roots of the equation
x² – 64x + 256 = 0. Then the value of
( (α ³)/(β ⁵) )^(1/8) + ( (β ³)/(α ⁵) )^(1/8) is :

Solution

Maths sir · 4 steps

Put both terms over the common denominator (αβ)^(5/8) and the numerator becomes α + β.

1. Sum and product. From x² - 64x + 256 = 0, α + β = 64 and αβ = 256. The discriminant 64² - 4 × 256 = 3072 is positive, and both the sum and the product are positive. So both roots are real and positive, and their eighth roots are real.
2. Common denominator. Multiply the top and bottom of the first term by α^(5/8). It becomes α^(3/8)α^(5/8)/(αβ)^(5/8) = α/(αβ)^(5/8). In the same way, the second term becomes β/(αβ)^(5/8).

   α^(3/8)/β^(5/8) + β^(3/8)/α^(5/8) = (α + β)/(αβ)^(5/8)

3. Put in numbers. 256 = 2⁸, so 256^(5/8) = 2⁵ = 32. The value is 64/32 = 2.
4. So. The expression equals 2, which is option (c).

The idea to keep. When the fractional powers of the two roots add up to 1, take a common denominator. The expression collapses into the sum and the product of the roots, and you never need the roots themselves.

Answer from the NTA final answer key. Solution by PrepWiser.