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

Given Equation: We are given the equation 2^(a) = root(9)(2^(4)). To solve for a, we need to express both sides of the equation with the same base and then equate the exponents.Rewriting Right Side: The cube root of a number is the same as raising that number to the power of 1/3. Therefore, we can rewrite the right side of the equation as (2^(4))^(1/3).Simplifying Exponents: Using the property of exponents that (x^(m))^(n) = x^(m*n), we can simplify the right side of the equation to 2^(4/3).Setting Exponents Equal: Now we have 2^(a) = 2^(4/3). Since the bases are the same, we can set the exponents equal to each other: a = 4/3.Final Answer: We have found the value of a to be 4/3. This is the final answer.

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Algebra 2

Factor sums and differences of cubes

Full solution

Q. If

2^{a}=\sqrt[9]{2^{4}}

, what is the value of

a

Given Equation: We are given the equation $2^{a} = \sqrt[9]{2^{4}}$ . To solve for $a$ , we need to express both sides of the equation with the same base and exponent format.
Express with Same Base: The ninth root of a number is the same as raising that number to the power of $\frac{1}{9}$ . Therefore, we can rewrite the equation as $2^{a} = (2^{4})^{\frac{1}{9}}$ .
Simplify Right Side: Using the property of exponents that $(x^{m})^{n} = x^{m*n}$ , we can simplify the right side of the equation to $2^{\frac{4}{9}}$ .
Equate Exponents: Now we have $2^{a} = 2^{\frac{4}{9}}$ . Since the bases are the same, we can equate the exponents: $\newline$ $a = \frac{4}{9}$
Final Answer: We have found the value of $a$ to be $\frac{4}{9}$ , which is the final answer.