Question

If [√(12)³]^5/2=(√2)^2a+1, find the value of a​

Answer 1

Step-by-step explanation:

Given:-

[√(12)³]^5/2=(√2)^2a+1

To find:-

Find the value of a?

Solution:-

Given equation is [√(12)³]^5/2=(√2)^2a+1

=>[ {(12)³}½]^5/2 = [(2)½]^(2a+1)

Since √a = a½

=> (12)^(3×1/2×5/2) = 2^(1/2×(2a+1))

Since (a^m)^n = a^(mn)

=> 12^(15/4) = 2^(2a+1)/2

12 can be written as 12 = 2×2×3 = 2²×3

=> (2²×3)^(15/4) = 2^(2a+1)/2

=>( 2²)^15/4)×(3)^(15/4) = 2^(2a+1)/2

Since (ab)^m =a^m × b^m

=> 2^(2×15/4)×3^(15/4) = 2^(2a+1)/2

=>2^(15/2)×3^(15/4) = 2^(2a+1)/2

=> 3^(15/4) = [2^(2a+1)/2]/[2^(15/2)]

=> 3^(15/4) = 2^[{(2a+1)/2}-(15/2)]

Since a^m / a^n = a^(m-n)

=> 3^(15/4) = 2^(2a+1-15)/2

=>3^(15/4) = 2^(2a-14)/2

=> 3^(15/4) = 2^[2(a-7)/2]

=>3^(15/4) = 2^(a-7)

On taking logarithms both sides then

=> log 3^(15/4) = log 2^(a-7)

=> (15/4) log 3 = (a-7) log 2

Since log a^m = m log a

=> (15/4) log 3 = a log 2 - 7 log 2

=> a log 2 - 7 log 2 = (15/4) log 3

=> a log 2 = (15/4) log 3 +7 log 2

=> a =[ (15/4) log 3 + 7 log 2 ]/ ( log 2)

=> a = [(15 log 3 + 28 log 2)/4]/(log 2)

=> a = (15 log 3 + 28 log 2)/(4 log 2)

Answer :-

The value of a for the given problem is

(15 log 3 + 28 log 2)/(4 log 2)

Used formulae:-

• √a = a½
• (ab)^m =a^m × b^m
• a^m / a^n = a^(m-n)
• log a^m = m log a
• (a^m)^n = a^(mn)