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If 11^(a)=root(4)(11^(3)), what is the value of a ?

If 11a=1134 11^{a}=\sqrt[4]{11^{3}} , what is the value of a a ?

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Full solution

Q. If 11a=1134 11^{a}=\sqrt[4]{11^{3}} , what is the value of a a ?
  1. Given Equation: We are given the equation 11a=113411^{a} = \sqrt[4]{11^{3}}. To solve for aa, we need to express both sides of the equation with the same base and exponent format.
  2. Express with Same Base: The fourth root of a number is the same as raising that number to the power of 14\frac{1}{4}. Therefore, we can rewrite the equation as 11a=(113)1411^{a} = (11^{3})^{\frac{1}{4}}.
  3. Simplify Right Side: Using the property of exponents that (xm)n=xm∗n(x^{m})^{n} = x^{m*n}, we can simplify the right side of the equation to 113411^{\frac{3}{4}}.
  4. Equate Exponents: Now we have 11a=113411^{a} = 11^{\frac{3}{4}}. Since the bases are the same, we can equate the exponents: \newlinea=34a = \frac{3}{4}
  5. Final Answer: We have found the value of aa to be 34\frac{3}{4}, which is the final answer.

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