Question

What is the following simplified product? Assume x greater-than-or-equal-to 0

2 StartRoot 8 x cubed EndRoot (3 StartRoot 10 x Superscript 4 Baseline EndRoot minus x StartRoot 5 x squared EndRoot)
24 x cubed StartRoot 5 x EndRoot minus 4 x squared StartRoot 10 x EndRoot
24 x cubed StartRoot 5 x EndRoot minus 4 x cubed StartRoot 10 x EndRoot
24 x cubed StartRoot 5 EndRoot minus 4 x cubed StartRoot 10 x EndRoot
24 x cubed StartRoot 5 x EndRoot + 4 x cubed StartRoot 10 x EndRoot

Answer 1

First simplify the radicals (x ≥ 0 so √(x^n) = x^{n/2}):

√(8x^3) = 2√2 · x^{3/2},
√(10x^4) = √10 · x^2,
√(5x^2) = √5 · x.

So
2√(8x^3)(3√(10x^4) − x√(5x^2))
= 2·(2√2 x^{3/2})·(3√10 x^2 − √5 x^2)
= 4√2 x^{7/2}(3√10 − √5).

Compute √2·√10 = 2√5 and √2·√5 = √10:
= 4 x^{7/2}(6√5 − √10)
= 24 x^{7/2}√5 − 4 x^{7/2}√10
= 24 x^3√(5x) − 4 x^3√(10x).

So the correct choice is 24 x^3√(5x) − 4 x^3√(10x).