MHB Factoring With Negative Powers

[mathdad]

Factor

(x^2 + 1)^(-2/3) + (x^2 + 1)^(-5/3)

Solution:

(x^2 + 1)^(-2/3)[1 + (x^2 + 1)^(2/5)]

Yes?

 

[MarkFL]

We are given to factor:

$$\left(x^2+1\right)^{-\frac{2}{3}}+\left(x^2+1\right)^{-\frac{5}{3}}$$

So, we factor out the expression with the smaller exponent, observing that $$-\frac{5}{3}<-\frac{2}{3}$$...and then we subtract that exponent:

$$\left(x^2+1\right)^{-\frac{5}{3}}\left(\left(x^2+1\right)^{-\frac{2}{3}-\left(-\frac{5}{3}\right)}+\left(x^2+1\right)^{-\frac{5}{3}-\left(-\frac{5}{3}\right)}\right)$$

Now, simplify the exponents:

$$\left(x^2+1\right)^{-\frac{5}{3}}\left(\left(x^2+1\right)^{\frac{3}{3}}+\left(x^2+1\right)^{0}\right)$$

$$\left(x^2+1\right)^{-\frac{5}{3}}\left(\left(x^2+1\right)+1\right)$$

$$\left(x^2+1\right)^{-\frac{5}{3}}\left(x^2+2\right)$$

 

[mathdad]

I selected the wrong smallest power.