How do I factor and simplify $-45x^3 (9x^2 + 3)^{-3/2} + 10x (9x^2 + 3)^{-1/2}$?

Question

I just found the derivative of a function, and now I am trying to factor and simplify the expression below. I have no idea how to factor terms with fractions as exponents especially when these fractions are expressed as a negative exponent. Could someone help me?

The expression is:$$-45x^3 (9x^2 + 3)^{-3/2} + 10x (9x^2 + 3)^{-1/2}$$

It would be great if someone could explain in detail "clearly" how to go about factoring an expression with negative fractions as exponents. In other words, I need clarification on the process instead of the answer.

Answer 1

$$-45x^3 (9x^2 + 3)^{-3/2} + 10x (9x^2 + 3)^{-1/2}=$$

$$=5x(-9x^2) (9x^2 + 3)^{-1/2-1} + 5x\cdot2 (9x^2 + 3)^{-1/2}=$$

$$=5x(-9x^2) (9x^2 + 3)^{-1/2}(9x^2 + 3)^{-1} + 5x\cdot2 (9x^2 + 3)^{-1/2}=$$

$$=5x(9x^2 + 3)^{-1/2}(-9x^2(9x^2 + 3)^{-1} + 2)=$$

$$=5x(9x^2 + 3)^{-1/2}(2-9x^2(9x^2 + 3)^{-1})$$

Answer 2

Hint: $(9x^2 + 3)^{-1/2}=(9x^2 + 3) \cdot (9x^2 + 3)^{-3/2}$

Answer 3

Hint: consider the two terms as fractions (negative exponent means that this factor belongs to the denominator...)... make sure the denominator in the two terms is $(9x^2+3)^{3/2}$ Then factorize the fraction...