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If 11^(a)=root(6)(11^(5)), what is the value of a ?

If 11a=1156 11^{a}=\sqrt[6]{11^{5}} , what is the value of a a ?

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Full solution

Q. If 11a=1156 11^{a}=\sqrt[6]{11^{5}} , what is the value of a a ?
  1. Given Equation: We are given the equation 11a=115611^{a} = \sqrt[6]{11^{5}}. 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 sixth root of a number is the same as raising that number to the power of 16\frac{1}{6}. Therefore, we can rewrite the equation as 11a=(115)1611^{a} = (11^{5})^{\frac{1}{6}}.
  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 115611^{\frac{5}{6}}.
  4. Equate Exponents: Now we have 11a=115611^{a} = 11^{\frac{5}{6}}. Since the bases are the same, we can equate the exponents: \newlinea=56a = \frac{5}{6}
  5. Final Answer: We have found the value of aa to be 56\frac{5}{6}, which is the final answer.

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