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

Given equation: We are given the equation 7^(a) = root(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 nth 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))^(1/8) = 7^(3 * 1/8) = 7^(3/8).Setting the exponents equal: Now the equation is 7^(a) = 7^(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 = 3/8.

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Math Problems

Grade 8

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: $a = \frac{3}{8}$ .