Question

simplify. express your answer using exponent ((y^{4}z^{-4})^{-12})

Explanation:

Step1: Apply power of a product rule

For \((a^m b^n)^p\), we use the rule \((a^m b^n)^p=a^{m\times p}b^{n\times p}\). Here \(a = y\), \(m = 4\), \(b=z\), \(n=- 4\) and \(p=-12\). So we have:
\((y^{4}z^{-4})^{-12}=y^{4\times(-12)}z^{-4\times(-12)}\)

Step2: Calculate the exponents

Calculate \(4\times(-12)\) for the exponent of \(y\): \(4\times(-12)=- 48\)
Calculate \(-4\times(-12)\) for the exponent of \(z\): \(-4\times(-12) = 48\)

So the simplified form is \(y^{-48}z^{48}\) or we can also write it as \(\frac{z^{48}}{y^{48}}\) (using the rule \(a^{-n}=\frac{1}{a^{n}}\)) but since the problem says to express using exponents, \(y^{-48}z^{48}\) is also correct. If we want positive exponents for all, it is \(\frac{z^{48}}{y^{48}}\)

Answer:

\(\frac{z^{48}}{y^{48}}\) (or \(y^{-48}z^{48}\))