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:

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

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

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

Now, simplify the exponents:

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

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

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

 

[mathdad]

I selected the wrong smallest power.