8. The solution for (125)^2n/3. X (27)^-
n/6/ (75)^-n/6
Answers
Given :- The solution for (125)^2n/3 * (27)^-n/6 / (75)^-n/6 ?
Solution :-
→ { (125)^(2n/3) * (27)^(-n/6) } / (75)^(-n/6)
→ { (5³)^(2n/3) * (3³)^(-n/6) } / (75)^(-n/6)
using (a^m)^n = (a)^(m * n) in numerator now,
→ [5^{3 * (2n/3)} * 3^{3 * (-n/6)}] / (75)^(-n/6)
→ [5^(2n) * 3^(-n/2)] / (75)^(-n/6)
now, breaking the denominator part ,
→ [5^(2n) * 3^(-n/2)] / (25 * 3)^(-n/6)
→ [5^(2n) * 3^(-n/2)] / (5² * 3)^(-n/6)
using (a * b)^m = a^m * b^m in denominator, we get,
→ [5^(2n) * 3^(-n/2)] / [(5²)^(-n/6) * 3^(-n/6)]
→ [5^(2n) * 3^(-n/2)] / [5^{2 * (-n/6)} * 3^(-n/6)]
→ [5^(2n) * 3^(-n/2)] / [5^(-n/3) * 3^(-n/6)]
now, using a^m / a^n = a^(m - n) in both terms we get,
→ 5^{2n - (-n/3)} * 3^{(-n/2) - (-n/6)}
→ 5^{(6n + n)/3} * 3^{(-3n + n)/6}
→ 5^(7n/3) * 3^(-2n/6)
→ 5^(7n/3) * 3^(-n/3)
finally using a^(-m) = 1/a^m , we get,
→ [ 5^(7n/3) / 3^(n/3) ] (Ans.)
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Given :- The solution for (125)^2n/3 * (27)^-n/6 / (75)^-n/6 ?
Solution :-
→ { (125)^(2n/3) * (27)^(-n/6) } / (75)^(-n/6)
→ { (5³)^(2n/3) * (3³)^(-n/6) } / (75)^(-n/6)
using (a^m)^n = (a)^(m * n) in numerator now,
→ [5^{3 * (2n/3)} * 3^{3 * (-n/6)}] / (75)^(-n/6)
→ [5^(2n) * 3^(-n/2)] / (75)^(-n/6)
now, breaking the denominator part ,
→ [5^(2n) * 3^(-n/2)] / (25 * 3)^(-n/6)
→ [5^(2n) * 3^(-n/2)] / (5² * 3)^(-n/6)
using (a * b)^m = a^m * b^m in denominator, we get,
→ [5^(2n) * 3^(-n/2)] / [(5²)^(-n/6) * 3^(-n/6)]
→ [5^(2n) * 3^(-n/2)] / [5^{2 * (-n/6)} * 3^(-n/6)]
→ [5^(2n) * 3^(-n/2)] / [5^(-n/3) * 3^(-n/6)]
now, using a^m / a^n = a^(m - n) in both terms we get,
→ 5^{2n - (-n/3)} * 3^{(-n/2) - (-n/6)}
→ 5^{(6n + n)/3} * 3^{(-3n + n)/6}
→ 5^(7n/3) * 3^(-2n/6)
→ 5^(7n/3) * 3^(-n/3)
finally using a^(-m) = 1/a^m , we get,
→ [ 5^(7n/3) / 3^(n/3) ] (Ans.)