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

`10x^4√(6)+x^3√(30x)-10x^4√(3)-x^3√(15x)`

Explanation:

Remove perfect squares from under the radicals.

`(√(10x^4)-x√(5x^2))(2√(15x^4)+√(3x^3))\\\\=(√(10x^4))(2√(15x^4)) +(√(10x^4))(√(3x^3)) -(x√(5x^2))(2√(15x^4)) -(x√(5x^2))(√(3x^3))\\\\=2√(150x^8)+√(30x^7)-2x√(75x^6)-x√(15x^5)\\\\=\boxed{10x^4√(6)+x^3√(30x)-10x^4√(3)-x^3√(15x)}`

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The applicable rules of exponents are ...

(x^a)(x^b) = x^(a+b)

√(a^2) = a . . . . . . . for a > 0

(√a)(√b) = √(ab)