Question

simplify. express your answer using exponents.
$(x^{-12}y^{6}z^{0})^{-6}$

Explanation:

Step1: Apply exponent rule \((ab)^n = a^n b^n\)

We can distribute the outer exponent \(-6\) to each factor inside the parentheses. So we get \(x^{-12\times(-6)}y^{6\times(-6)}z^{0\times(-6)}\).

Step2: Simplify each exponent

• For the \(x\) term: \(-12\times(-6)=72\), so we have \(x^{72}\).
• For the \(y\) term: \(6\times(-6)= - 36\), so we have \(y^{-36}\). But we can also remember that \(a^{-n}=\frac{1}{a^{n}}\), but since we need to express with exponents (and negative exponents are allowed in the expression), or we can rewrite \(y^{-36}\) as \(\frac{1}{y^{36}}\), but let's check the \(z\) term.
• For the \(z\) term: \(z^{0\times(-6)} = z^{0}=1\) (since any non - zero number to the power of 0 is 1). So the \(z\) term disappears (multiplies by 1).

Now, combining the terms, we have \(x^{72}y^{-36}\times1=x^{72}y^{-36}\) or \(\frac{x^{72}}{y^{36}}\). But if we want to write with positive exponents for \(y\), it is \(\frac{x^{72}}{y^{36}}\), but the problem says "express your answer using exponents", and negative exponents are a valid way to express with exponents. However, usually, we can also write it as \(x^{72}y^{-36}\) or simplify further. Wait, let's re - check the exponent rules. The power of a power rule is \((a^{m})^{n}=a^{m\times n}\).

So \((x^{- 12}y^{6}z^{0})^{-6}=x^{-12\times(-6)}y^{6\times(-6)}z^{0\times(-6)}=x^{72}y^{-36}z^{0}\). Since \(z^{0} = 1\), the expression simplifies to \(x^{72}y^{-36}\) or \(\frac{x^{72}}{y^{36}}\).

Answer:

\(x^{72}y^{-36}\) (or \(\frac{x^{72}}{y^{36}}\))