Question

simplify. express your answer using positive exponents.
$p^{-6}q^{-45}r\cdot p^{0}q^{6}r^{7}\cdot p^{65}q^{-1}r^{9}$

Explanation:

Step1: Group like bases

Group terms with $p$, $q$, $r$ separately:
$(p^{-6} \cdot p^{0} \cdot p^{65}) \cdot (q^{-45} \cdot q^{6} \cdot q^{-1}) \cdot (r^{1} \cdot r^{7} \cdot r^{9})$

Step2: Apply exponent product rule

Add exponents for same bases: $a^m \cdot a^n = a^{m+n}$
For $p$: $p^{-6+0+65} = p^{59}$
For $q$: $q^{-45+6+(-1)} = q^{-40}$
For $r$: $r^{1+7+9} = r^{17}$

Step3: Convert negative exponent to positive

Use $a^{-n} = \frac{1}{a^n}$ for $q^{-40}$:
$\frac{1}{q^{40}}$

Step4: Combine all terms

Multiply the simplified terms together.

Answer:

$p^{59}r^{17}\frac{1}{q^{40}}$ or $\frac{p^{59}r^{17}}{q^{40}}$