Simplifying problem

In the equation

f(x) = -3( (2-3x^2)^-1/2 + 3x^2(2-3x^2)^-3/2)

my text book jumps to the simplification

f(x) = -6(2-3x^2)^-3/2)

Could someone please explain this simplification?

Hello Hefotos

Quote:

Originally Posted by Hefotos

In the equation

f(x) = -3( (2-3x^2)^-1/2 + 3x^2(2-3x^2)^-3/2)

my text book jumps to the simplification

f(x) = -6(2-3x^2)^-3/2)

Could someone please explain this simplification?

Note first that:

$(2-3x^2)^{-\frac12}=(2-3x^2)^{-\frac32}\times(2-3x^2)^1$
So we can take out $(2-3x^2)^{-\frac32}$ as a common factor:

$f(x)=-3\Big( (2-3x^2)^{-\frac12} + 3x^2(2-3x^2)^{-\frac32}\Big)$

$=-3(2-3x^2)^{-\frac32}\Big( (2-3x^2)^1 + 3x^2\Big)$

$=-3(2-3x^2)^{-\frac32}( 2)$

$=-6(2-3x^2)^{-\frac32}$

Grandad

Very well explained, thankyou