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If
2^(a)=root(7)(2^(5)), what is the value of
a ?

If $2^{a}=\sqrt[7]{2^{5}}$ , what is the value of $a$ ?

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Solve equations using cube roots

Full solution

Q. If

2^{a}=\sqrt[7]{2^{5}}

, what is the value of

a

?

Given Equation: We are given the equation $2^{a} = \sqrt[7]{2^{5}}$ . To solve for $a$ , we need to express both sides of the equation with the same base and then compare the exponents.
Express with Same Base: The $7^{th}$ root of $2^{5}$ can be written as $(2^{5})^{(1/7)}$ . This uses the property that the $n^{th}$ root of a number is the same as raising that number to the power of $1/n$ .
Simplify Right Side: Now we have $2^{a} = (2^{5})^{1/7}$ . Using the property of exponents that $(x^{m})^{n} = x^{m*n}$ , we can simplify the right side of the equation.
Simplify Exponents: Simplify the right side of the equation: $(2^{5})^{\frac{1}{7}} = 2^{5 \cdot \frac{1}{7}} = 2^{\frac{5}{7}}$ .
Set Exponents Equal: Now the equation is $2^{a} = 2^{\frac{5}{7}}$ . Since the bases are the same, we can set the exponents equal to each other. Therefore, $a = \frac{5}{7}$ .

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