Question

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

(StartRoot 10 x Superscript 4 Baseline EndRoot minus x StarRoot 5 x squared EndRoot) (2 StartRoot 15 x Superscript 4 Baseline EndRoot + StartRoot 3 x cubed EndRoot)
10 x Superscript 4 Baseline StartRoot 6 EndRoot + x cubed StartRoot 30 x EndRoot minus 10 x Superscript 4 Baseline StartRoot 3 EndRoot + x squared StartRoot 15 x EndRoot
11 x Superscript 4 Baseline StartRoot 6 EndRoot + x cubed StartRoot 30 x EndRoot minus x Superscript 4 Baseline StartRoot 75 EndRoot + x squared StartRoot 15 EndRoot
10 x Superscript 4 Baseline StartRoot 6 EndRoot + x cubed StartRoot 30 x EndRoot minus 10 x Superscript 4 Baseline StartRoot 3 EndRoot minus x squared StartRoot 15 EndRoot
11 x Superscript 4 Baseline StartRoot 6 EndRoot + x cubed StartRoot 30 x EndRoot minus 10 x Superscript 4 Baseline StartRoot 3 EndRoot minus x cubed StartRoot 15 x EndRoot

Answer 1

Assuming x ≥ 0, √(x^4) = x^2 and √(x^3) = x√x. So

(√(10x^4) − x√(5x^2))(2√(15x^4) + √(3x^3))
= x^2(√10 − √5)(2x^2√15 + x√(3x))
= 10x^4√6 − 10x^4√3 + x^3√(30x) − x^3√(15x).

Equivalently: 10x^4(√6 − √3) + x^3(√(30x) − √(15x)).