Simplify (x^a)^(-b), where a and b are constants.

Step-by-step solution

1. 1

   Apply the power-of-a-power rule. For any base $u$ and exponents $m,n$:

   $\left(u^m\right)^n=u^{mn}$

   The exponents multiply; a common slip is to add them, which is the rule for $u^m\cdot u^n$ instead.

2. 2

   Substitute the given exponents. With $u=x$, $m=a$, $n=-b$:

   $\left(x^a\right)^{-b}=x^{a\cdot(-b)}=x^{-ab}$

3. 3

   Convert the negative exponent to a reciprocal. By definition $u^{-k}=\tfrac{1}{u^k}$ for $u\neq 0$:

   $x^{-ab}=\frac{1}{x^{ab}}$

   Both forms are correct; the fraction form is usually preferred when 'simplify' means 'no negative exponents'.

4. 4

   State the domain restriction. The manipulation needs $x\neq 0$, and for arbitrary real exponents the standard convention is $x>0$ so that $x^a$ is defined. Under those conditions the identity holds exactly.

5. 5

   Check with numbers. Take $x=2$, $a=3$, $b=2$: the original is $\left(2^3\right)^{-2}=8^{-2}=\tfrac{1}{64}$, and $x^{-ab}=2^{-6}=\tfrac{1}{64}$ $\checkmark$. A non-integer test: $x=2.3$, $a=1.7$, $b=0.8$ gives $\left(2.3^{1.7}\right)^{-0.8}\approx 0.322144$ and $2.3^{-1.36}\approx 0.322144$ $\checkmark$.