How many digits does a number 8^n.5^3n+4 have, if ?
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Given : 8ⁿ.5³ⁿ⁺⁴
To Find : How many digits number have
Solution:
8ⁿ.5³ⁿ⁺⁴
xᵃ⁺ᵇ = xᵃ.xᵇ
= (2³)ⁿ.5³ⁿ.5⁴
(xᵃ)ᵇ = xᵃᵇ
=2³ⁿ.5³ⁿ.5⁴
xᵃ.yᵃ= (xy)ᵃ
= (2 * 5)³ⁿ.5⁴
= (10)³ⁿ.5⁴
5⁴ = 625
= 625(10)³ⁿ
Hence number of digits = 3n + 3
LETS CHECK
n = 0 Digits = 3
8ⁿ.5³ⁿ⁺⁴ = 8⁰.5⁴ = 625 3 digits
n = 2 Digits = 8
8².5⁷ = 8⁰.5⁴ = 50,00,000 8 digits
8ⁿ.5³ⁿ⁺⁴ has 3n + 3 digit
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