Answer:
[tex]\frac{1}{5}[/tex]
Step-by-step explanation:
Using the rules of exponents
[tex]a^{m}[/tex] × [tex]a^{n}[/tex] = [tex]a^{(m+n)}[/tex], [tex]\frac{a^{m} }{a^{n} }[/tex] = [tex]a^{(m-n)}[/tex], [tex](a^m)^{n}[/tex] = [tex]a^{mn}[/tex]
Simplifying the product of the first 2 terms
[tex]\frac{a^{p^2+pq} }{a^{pq+q^2} }[/tex] × [tex]\frac{a^{q^2+qr} }{a^{qr+r^2} }[/tex]
= [tex]a^{p^2-q^2}[/tex] × [tex]a^{q^2-r^2}[/tex]
= [tex]a^{p^2-r^2}[/tex]
Simplifying the third term
5([tex](a^p+r)^{p-r}[/tex]
= 5[tex]a^{(p+r)(p-r)}[/tex] = 5[tex]a^{(p^2-r^2)}[/tex]
Performing the division, that is
[tex]\frac{a^{(p^2-r^2)} }{5a^{(p^2-r^2)} }[/tex] ← cancel [tex]a^{(p^2-r^2)}[/tex] on numerator/ denominator leaves
= [tex]\frac{1}{5}[/tex]
