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

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Solve equations with variable exponents

Full solution

Q. If

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

, what is the value of

a

?

Identify Given Equation: Identify the given equation and the goal. $\newline$ We are given that $2^a = \sqrt[7]{2^5}$ , and we need to find the value of $a$ .
Express Seventh Root: Express the seventh root as an exponent. $\newline$ The seventh root of $2^5$ can be written as $(2^5)^{\frac{1}{7}}$ .
Apply Power Rule: Apply the power rule for exponents, which states that $(a^m)^n = a^{m*n}$ . $\newline$ So, $(2^5)^{1/7}$ becomes $2^{5*(1/7)}$ .
Multiply Exponents: Multiply the exponents to simplify the expression. $\newline$ $5 \times \left(\frac{1}{7}\right) = \frac{5}{7}$ . $\newline$ So, $2^{5\times\left(\frac{1}{7}\right)}$ simplifies to $2^{\frac{5}{7}}$ .
Set Equal to $2^a$ : Set the simplified expression equal to $2^a$ . We have $2^a = 2^{5/7}$ .
Determine Value of $a$ : Since the bases are the same, the exponents must be equal. $\newline$ Therefore, $a = \frac{5}{7}$ .

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