Question

simplify. express your answer using exponents.
$(y^{4}z^{9})^{-4}$

Explanation:

Step1: Apply power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So for \((y^{4}z^{9})^{-4}\), we can apply this rule:
\((y^{4}z^{9})^{-4}=(y^{4})^{-4}(z^{9})^{-4}\)

Step2: Apply power of a power rule

The power of a power rule states that \((a^m)^n=a^{m\times n}\). For \((y^{4})^{-4}\), we have \(m = 4\) and \(n=- 4\), so \(y^{4\times(-4)}=y^{-16}\). For \((z^{9})^{-4}\), we have \(m = 9\) and \(n =-4\), so \(z^{9\times(-4)}=z^{-36}\).
Combining these results, we get \(y^{-16}z^{-36}\). If we want to express this with positive exponents (though the problem just says to use exponents, negative exponents are also exponents), we can use the rule \(a^{-n}=\frac{1}{a^{n}}\), but since the problem doesn't specify positive exponents, \(y^{-16}z^{-36}\) is also a simplified form. However, usually, we can also write it as \(\frac{1}{y^{16}z^{36}}\) but let's check the steps again. Wait, the problem says "express your answer using exponents", negative exponents are exponents, so the first form is also correct.

Wait, let's redo the steps correctly. The power of a product rule: \((ab)^n=a^n b^n\), so \((y^{4}z^{9})^{-4}=y^{4\times(-4)}z^{9\times(-4)}\) (directly applying \((a^m b^n)^p=a^{m\times p}b^{n\times p}\), which is a combination of power of product and power of power). So \(4\times(-4)=- 16\) and \(9\times(-4)=-36\). So the simplified form is \(y^{-16}z^{-36}\) or \(\frac{1}{y^{16}z^{36}}\). But since the problem says "using exponents", negative exponents are exponents, so \(y^{-16}z^{-36}\) is acceptable.

Answer:

\(y^{-16}z^{-36}\) (or \(\frac{1}{y^{16}z^{36}}\))