Question

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

(4 x StartRoot 5 x squared EndRoot + 2 x squared StartRoot 6 EndRoot) squared
80 x Superscript 4 Baseline + 8 x Superscript 4 Baseline StartRoot 30 x EndRoot + 24 x Superscript 4
80 x Superscript 6 Baseline + 8 x Superscript 5 Baseline + 8 x Superscript 5 Baseline StartRoot 30 EndRoot + 24 x Superscript 4
104 x Superscript 4
104 x Superscript 4 + 16 x Superscript 4 Baseline StartRoot 30 EndRoot

Answer 1

Start with
(4x√(5x^2) + 2x^2√6)^2.

Since x ≥ 0, √(5x^2) = x√5, so the expression inside is
4x·(x√5) + 2x^2√6 = 4x^2√5 + 2x^2√6 = x^2(4√5 + 2√6).

Squaring gives
x^4(4√5 + 2√6)^2 = x^4( (4√5)^2 + 2·(4√5)(2√6) + (2√6)^2 )
= x^4(80 + 16√30 + 24)
= x^4(104 + 16√30).

You can also write it as 8x^4(13 + 2√30).