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If 11a=411311^a = 4\sqrt{11^3}, what is the value of aa?

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Full solution

Q. If 11a=411311^a = 4\sqrt{11^3}, what is the value of aa?
  1. Identify Equation & Operation: Identify the given equation and the mathematical operation involved.\newlineThe given equation is 11a=(113)1411^a = (11^3)^{\frac{1}{4}}.
  2. Rewrite Using Exponent Property: Rewrite the right side of the equation using the property of exponents that states (am)1/n=am/n(a^m)^{1/n} = a^{m/n}.\newlineSo, (113)1/4(11^3)^{1/4} becomes 113/411^{3/4}.
  3. Set Exponents Equal: Set the exponents equal to each other since the bases are the same.\newlineTherefore, a=34a = \frac{3}{4}.
  4. Check Solution: Check the solution by substituting aa back into the original equation.113411^{\frac{3}{4}} should equal the fourth root of 11311^3. Since 113411^{\frac{3}{4}} is indeed the fourth root of 11311^3, the solution is correct.

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