Question

simplify. express your answer using exponents.
$(q^{-11}r^{11})^2$

Explanation:

Step1: Apply power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, for \((q^{-11}r^{11})^2\), we can apply this rule:
\((q^{-11})^2 \cdot (r^{11})^2\)

Step2: Apply power of a power rule

The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). Applying this to each term:
For \((q^{-11})^2\), we have \(q^{-11 \cdot 2}=q^{-22}\)
For \((r^{11})^2\), we have \(r^{11 \cdot 2}=r^{22}\)

Step3: Combine the terms

Putting the two results together, we get \(q^{-22}r^{22}\) (or it can also be written as \(\frac{r^{22}}{q^{22}}\) using the negative exponent rule \(a^{-n}=\frac{1}{a^n}\), but since the problem just asks to express using exponents, \(q^{-22}r^{22}\) is also acceptable. However, if we want to write with positive exponents for the \(q\) term, we can rearrange as \(\frac{r^{22}}{q^{22}}\) or keep it as \(q^{-22}r^{22}\). But typically, when simplifying with exponents, both forms are okay, but let's check the steps again. Wait, actually, when we apply the power of a product and power of a power, we get \(q^{-22}r^{22}\) which can be written as \(r^{22}q^{-22}\) or \(\frac{r^{22}}{q^{22}}\). But let's confirm the exponent rules.

Power of a product: \((ab)^n = a^n b^n\) – correct.

Power of a power: \((a^m)^n = a^{mn}\) – correct. So \((q^{-11})^2 = q^{-22}\), \((r^{11})^2 = r^{22}\). So multiplying them gives \(q^{-22}r^{22}\) or \(\frac{r^{22}}{q^{22}}\). But the problem says "express your answer using exponents" – both forms use exponents, but maybe the first form is acceptable. Alternatively, we can write it as \(r^{22}q^{-22}\) or \(\frac{r^{22}}{q^{22}}\). But let's see, the original expression is \((q^{-11}r^{11})^2\), so applying the rules:

\((q^{-11}r^{11})^2 = (q^{-11})^2 \times (r^{11})^2 = q^{-22} \times r^{22} = q^{-22}r^{22}\) (or \(r^{22}q^{-22}\) or \(\frac{r^{22}}{q^{22}}\)).

Answer:

\(q^{-22}r^{22}\) (or \(\frac{r^{22}}{q^{22}}\))