Question

simplify. express your answer using exponents.
$(q^{4}r^{-1})^{-9}$

Explanation:

Step1: Apply the power of a product rule

We use the rule \((ab)^n = a^n b^n\), so \((q^{4}r^{-1})^{-9}=q^{4\times(-9)}r^{-1\times(-9)}\)

Step2: Simplify the exponents

Calculate \(4\times(-9)= - 36\) and \(-1\times(-9) = 9\), so we get \(q^{-36}r^{9}\) or we can write it as \(\frac{r^{9}}{q^{36}}\) (using the rule \(a^{-n}=\frac{1}{a^{n}}\)), but since the problem asks to express the answer using exponents, \(q^{-36}r^{9}\) is also acceptable. However, usually we prefer positive exponents in the numerator, so \(\frac{r^{9}}{q^{36}}\) or \(q^{-36}r^{9}\). But if we follow the form with exponents as they are after simplifying the multiplication, we have:

After step 2, the exponents are calculated, so the simplified form is \(q^{-36}r^{9}\) (or \(\frac{r^{9}}{q^{36}}\))

Answer:

\(q^{-36}r^{9}\) (or \(\frac{r^{9}}{q^{36}}\))