Answer:
Vales of a =3 and b = 6
Step-by-step explanation:
Given that: [tex](ar^b)^4 = 81r^{24}[/tex] .....[1] where a and b are positive integers
we can write 81 and 24 as;
[tex]81 = 3 \cdot 3 \cdot 3 \cdot 3 = 3^4[/tex]
[tex]24 = 4 \cdot 6[/tex]
We have [1] as;
[tex](ar^b)^4 = 3^4r^{4 \cdot 6}[/tex]
Using power rules;
[tex]a^nb^n = (ab)^n[/tex]
[tex]a^n = b^n[/tex] which implies a = b
then;
[tex](ar^b)^4 = (3r^6)^4[/tex]
[tex]ar^b = 3r^6[/tex]
On comparing both sides we have;
a = 3 and
[tex]r^b = r^6[/tex]
⇒[tex]b = 6[/tex]
Therefore, the value of a and b are, 3 and 6
