Question

3^a=81^b+2 and 125^b=5^a_3 find ab

Answer 1

3^a=3^(4(b+2))
(Base are same so)
a=4b+2

125^b=5^3a
5^3b=5^3a
(Base are same so)
3b=3a
a=b
so,
a=4b+2
b=4b+2
b=-(3/2)
a=b=-(3/2)
ab=9/4

Hope it helps.
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Answer 2

Answer:

Hey Mate!!

$3^{a} = 81^{b+2}\\=> 3^{a} = 3^{4(b+2)}\\=> 3^{a} = 3^{4b+8}\\=> a = 4b + 8 \:\:\:\:\:.....[Bases\: are\: same,\: powers \: are\: equal]\\\\125^{b} = 5^{a-3} \\=> 5^{3b} = 5^{a-3}\\=> 3b = a-3\:\:\:\:\:.....[Bases\: are\: same,\: powers \: are\: equal]\\\\On\: substituting \: a,\\3b = 4b + 8-3\\=> 3b-4b=5\\=>-b=5\\=> \bold{b=(-5)}\\\\On\: substituting \: b\; in\: first\: equation,\\a = 4b+8\\=>a=4(-5)+8\\=>a=-20+8\\=>\bold{a=(-12)}\\\\\boxed{\bold{Therefore,\:ab = (-12)(-5)=60}}$