Factor and simplify the algebraic expression: (x + 6) to the power negative one-fifth minus (x + 6) to the power negative six-fifths.

Step-by-step solution

1. 1

   Choose the common factor. When two powers of the same base are subtracted, the factor to take out is the base raised to the smaller exponent. Comparing $-\frac15$ and $-\frac65$, the smaller is $-\frac65$.

2. 2

   Divide each term by that factor. $(x+6)^{-1/5} \div (x+6)^{-6/5} = (x+6)^{-1/5 + 6/5} = (x+6)^{1}$, and the second term divides to $1$.

3. 3

   Write the factored form. $(x+6)^{-6/5}\left[(x+6) - 1\right] = (x+6)^{-6/5}(x+5)$.

4. 4

   Clear the negative exponent. A negative exponent means a reciprocal, so the result becomes $\dfrac{x+5}{(x+6)^{6/5}}$.

5. 5

   Check numerically. At $x = 26$: $(32)^{-1/5} - (32)^{-6/5} = 0.5 - 0.015625 = 0.484375$, and $\dfrac{31}{32^{6/5}} = \dfrac{31}{64} = 0.484375$.