The sqrt of 3(8^sqrt of 7 - 4^sqrt of 3)
To simplify the given expression, let's break it down step by step.
First, let's simplify the exponents by applying the basic exponent rules:
√(3(8^√7 - 4^√3))
= √(3(8^√7/4^√3)) [Using the rule: a^(m-n) = a^m / a^n]
= √(3((2^3)^√7/(2^2)^√3)) [Expressing 8 and 4 as powers of 2]
= √(3(2^(3√7)/(2^(2√3))))
= √(3 * (2^(√7 * 3)/(2^(√3 * 2)))) [Using the rule: (a^m)^n = a^(m * n)]
= √(3 * (2^(3√7 - 2√3))) [Using the rule: a^m / a^n = a^(m-n)]
Now, let's simplify the expression inside the square root:
= √(3 * 2^(3√7 - 2√3))
Since we cannot simplify any further, this is the simplified expression for √(3(8^√7 - 4^√3)).