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If
7^(a)=root(8)(7^(3)), what is the value of
a ?

If $7^{a}=\sqrt[8]{7^{3}}$ , what is the value of $a$ ?

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Solve equations using cube roots

Full solution

Q. If

7^{a}=\sqrt[8]{7^{3}}

, what is the value of

a

?

Given equation: We are given the equation $7^{a} = \sqrt[8]{7^{3}}$ . To solve for $a$ , we need to express both sides of the equation with the same base and exponent format.
Expressing both sides: The eighth root of $7^{3}$ can be written as $(7^{3})^{1/8}$ . This uses the property that the $n$ th root of a number is the same as raising that number to the power of $1/n$ .
Using the property of exponents: Now we have $7^{a} = (7^{3})^{1/8}$ . Using the property of exponents that $(x^{m})^{n} = x^{m*n}$ , we can simplify the right side of the equation.
Simplifying the right side: Simplify the right side of the equation: $(7^{3})^{\frac{1}{8}} = 7^{3 \times \frac{1}{8}} = 7^{\frac{3}{8}}$ .
Setting the exponents equal: Now the equation is $7^{a} = 7^{\frac{3}{8}}$ . Since the bases are the same and the expressions are equal, the exponents must also be equal.
Setting the exponents equal: Now the equation is $7^{a} = 7^{\frac{3}{8}}$ . Since the bases are the same and the expressions are equal, the exponents must also be equal.Set the exponents equal to each other: $a = \frac{3}{8}$ .

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