Factor and simplify the algebraic expression (x + 6)^(-1/5) - (x + 6)^(-6/5).

Step-by-step solution

1. 1

   Compare the two exponents on the number line. The exponents are $-\tfrac15$ and $-\tfrac65$, and $-\tfrac65<-\tfrac15$, so the smaller one is $-\tfrac65$. The rule is the same as with positive exponents: factor out the base raised to the smallest exponent, which here is the most negative one.

2. 2

   Divide each term by that common factor. The first term gives $(x+6)^{-1/5-(-6/5)}=(x+6)^{5/5}=x+6$, and the second gives $(x+6)^{0}=1$, so $(x+6)^{-1/5}-(x+6)^{-6/5}=(x+6)^{-6/5}\bigl[(x+6)-1\bigr].$

3. 3

   Simplify the bracket. $(x+6)-1=x+5 .$

4. 4

   Rewrite with a positive exponent. Since $(x+6)^{-6/5}=\dfrac{1}{(x+6)^{6/5}}$, $(x+6)^{-1/5}-(x+6)^{-6/5}=\frac{x+5}{(x+6)^{6/5}} .$

5. 5

   Note the domain and check a value. The expression needs $x+6>0$, i.e. $x>-6$, for the fractional powers to be defined and nonzero. At $x=-5$ the original is $1^{-1/5}-1^{-6/5}=0$, and the answer gives $\frac{0}{1}=0$, so the two agree.