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If 3^(a)=root(5)(3^(4)), what is the value of a ?

If 3a=345 3^{a}=\sqrt[5]{3^{4}} , what is the value of a a ?

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Full solution

Q. If 3a=345 3^{a}=\sqrt[5]{3^{4}} , what is the value of a a ?
  1. Given Information: We are given that 3a3^{a} is equal to the fifth root of 343^{4}. The fifth root of a number is the same as raising that number to the power of 15\frac{1}{5}. Therefore, we can rewrite the equation as:\newline3a=(34)153^{a} = (3^{4})^{\frac{1}{5}}
  2. Rewriting the Equation: 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:\newline3a=34⋅153^{a} = 3^{4 \cdot \frac{1}{5}}
  3. Simplifying the Equation: Now we have 3a=3453^{a} = 3^{\frac{4}{5}}. Since the bases are the same and the equation states that the two expressions are equal, we can set the exponents equal to each other:\newlinea=45a = \frac{4}{5}
  4. Setting the Exponents Equal: We have found the value of aa to be 45\frac{4}{5}. There is no need for further calculation or simplification.

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