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If 2^(a)=root(9)(2^(4)), what is the value of a ? ◻

If 2a=2492^{a}=\sqrt[9]{2^{4}}, what is the value of aa ?\newline◻

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Csc, sec, and cot of special anglesright chevron icon

Full solution

Q. If 2a=2492^{a}=\sqrt[9]{2^{4}}, what is the value of aa ?\newline◻
  1. Rewrite Equation: Understand the given equation and rewrite it in a more familiar form.\newlineThe ninth root of 242^{4} can be written as (24)19(2^{4})^{\frac{1}{9}}.\newlineSo, the equation becomes 2a=(24)192^{a} = (2^{4})^{\frac{1}{9}}.
  2. Apply Power Rule: Apply the power rule of exponents which states that (xm)n=xm∗n(x^{m})^{n} = x^{m*n}.\newlineSo, (24)19(2^{4})^{\frac{1}{9}} becomes 24∗192^{4*\frac{1}{9}}.
  3. Simplify Exponents: Multiply the exponents to simplify the right side of the equation.\newline4×(19)=494\times\left(\frac{1}{9}\right) = \frac{4}{9}.\newlineSo, 2a=2492^{a} = 2^{\frac{4}{9}}.
  4. Solve for aa: Since the bases are the same and the equation is an equality, the exponents must be equal.\newlineTherefore, a=49a = \frac{4}{9}.

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