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)

Answer 1

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

√(10x^4) = x^2√10, x·√(5x^2) = x^2√5,
2√(15x^4) = 2x^2√15, √(3x^3) = x^{3/2}√3.

So the product is
(x^2(√10 − √5)) (2x^2√15 + x^{3/2}√3)
= (√10 − √5)(2√15 x^4 + √3 x^{7/2}).

Expanding and combining gives
10√6 x^4 − 10√3 x^4 + √30 x^{7/2} − √15 x^{7/2},

or compactly
10 x^4(√6 − √3) + x^{7/2}(√30 − √15).