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If
3^(a)=root(5)(3^(2)), what is the value of
a ?

If $3^{a}=\sqrt[5]{3^{2}}$ , what is the value of $a$ ?

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Find derivatives using the chain rule II

Full solution

Q. If

3^{a}=\sqrt[5]{3^{2}}

, what is the value of

a

?

Given Equation: We are given the equation $3^{a} = \sqrt[5]{3^{2}}$ . 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 fifth root of $3^{2}$ can be written as $(3^{2})^{1/5}$ . 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 $3^{a} = (3^{2})^{1/5}$ . Using the property of exponents that $(x^{m})^{n} = x^{m*n}$ , we can simplify the right side of the equation.
Set Exponents Equal: Simplify the right side: $(3^{2})^{\frac{1}{5}} = 3^{2\cdot\frac{1}{5}} = 3^{\frac{2}{5}}$ .
Set Exponents Equal: Simplify the right side: $(3^{2})^{1/5} = 3^{2*1/5} = 3^{2/5}$ .Now we have $3^{a} = 3^{2/5}$ . Since the bases are the same, we can set the exponents equal to each other to find the value of $a$ .
Set Exponents Equal: Simplify the right side: $(3^{2})^{\frac{1}{5}} = 3^{2*\frac{1}{5}} = 3^{\frac{2}{5}}$ .Now we have $3^{a} = 3^{\frac{2}{5}}$ . Since the bases are the same, we can set the exponents equal to each other to find the value of $a$ .Set the exponents equal: $a = \frac{2}{5}$ .

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