Question

simplify. express your answer using exponents.
$(w^{12}x^{-3})^{-7}$

Explanation:

Step1: Apply power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So for \((w^{12}x^{-3})^{-7}\), we apply this rule:
\(w^{12\times(-7)}x^{-3\times(-7)}\)

Step2: Calculate the exponents

First, calculate \(12\times(-7)\): \(12\times(-7)= - 84\)
Then, calculate \(-3\times(-7)\): \(-3\times(-7)=21\)
So we get \(w^{-84}x^{21}\)
But we can also write this with positive exponents for the \(w\) term using the rule \(a^{-n}=\frac{1}{a^{n}}\), but since the problem just says to express using exponents (it doesn't specify positive), \(w^{-84}x^{21}\) is also correct. However, if we want positive exponents for all, it would be \(\frac{x^{21}}{w^{84}}\), but let's check the original operation. Wait, the power of a power rule is \((a^m)^n=a^{m\times n}\), so we used that correctly. So the simplified form is \(w^{-84}x^{21}\) or \(\frac{x^{21}}{w^{84}}\). But let's follow the steps again.

Wait, the expression is \((w^{12}x^{-3})^{-7}\). Using the power of a product rule (which is an extension of the power of a power rule for products), we multiply each exponent inside the parentheses by the exponent outside. So for \(w^{12}\), the new exponent is \(12\times(-7)=-84\), and for \(x^{-3}\), the new exponent is \(-3\times(-7) = 21\). So the simplified expression is \(w^{-84}x^{21}\). If we want to write with positive exponents, we can use \(a^{-n}=\frac{1}{a^{n}}\), so \(w^{-84}=\frac{1}{w^{84}}\), so the expression becomes \(\frac{x^{21}}{w^{84}}\). But the problem says "express your answer using exponents", and negative exponents are still exponents, so both are correct, but maybe the expected form is with the exponents as calculated.

Answer:

\(w^{-84}x^{21}\) (or \(\frac{x^{21}}{w^{84}}\))