MHB Simplifying [(x2 - 4)(x2 + 3)1/2] - [(x2 - 4)2(x2 + 3)3/2]

[PaperStSoap]

[(x² - 4)(x² + 3)^1/2] - [(x² - 4)²(x² + 3)^3/2]

I understand the first half which comes out to...

(x-2)(x+2)(x² + 3)^1/2 I have know idea how to start the second half.

Thanks in advance.

 

[CaptainBlack]

[(x² - 4)(x² + 3)^1/2] - [(x² - 4)²(x² + 3)^3/2]

I understand the first half which comes out to...

(x-2)(x+2)(x² + 3)^1/2 I have know idea how to start the second half.

Thanks in advance.

Look for the greatest common factor of the two terms, in this case that is \(x^2-4)(x^2+3)^{1/2}\), so:
\[\begin{aligned}\left((x^2 - 4)(x^2 + 3)^{1/2}\right) - \left((x^2 - 4)^2(x^2 + 3)^{3/2}\right)&=(x^2-4)(x^2+3)^{1/2}\left(1-(x^2-4)(x^2+3) \right)\\
&=(4-x^2)(x^2+3)^{1/2}\left((x^2-4)(x^2+3)-1 \right) \end{aligned}\]

CB

 

[soroban]

Hello, PaperStSoap!

\text{Simplify: }\: (x^2-4)(x^2+3)^{\frac{1}{2}} - (x^2-4)^2(x^2+3)^{\frac{3}{2}}

Factor: .$(x^2-4)(x^2+3)^{\frac{1}{2}}\cdot\big[1 - (x^2-4)(x^2+3)\big] $

. . . . $=\;(x^2-4)(x^2+3)^{\frac{1}{2}}\cdot\big[1 - x^4 + x^2 + 12\big]$

. . . . $=\;(x^2-4)(x^2+3)^{\frac{1}{2}}\left(13 + x^2 - x^4\right)$