Which is equivalent to RootIndex 3 StartRoot 8 EndRoot Superscript x? RootIndex x StartRoot 8 EndRoot Superscript x 8 Superscript StartFraction 3 Over x EndFraction 8 Superscript StartFraction x Over 3 EndFraction 8 Superscript 3 x
The Correct Answer and Explanation is:
The given expression is:
[
\sqrt[3]{8^x}
]
We need to simplify this and express it in a form that matches one of the provided options.
Step 1: Rewriting the expression
The cube root of ( 8^x ) can be rewritten using exponent rules:
[
\sqrt[3]{8^x} = (8^x)^{\frac{1}{3}}
]
By the power of a power rule ( (a^m)^n = a^{m \cdot n} ), we can multiply the exponents:
[
(8^x)^{\frac{1}{3}} = 8^{\frac{x}{3}}
]
Step 2: Expressing ( 8^{\frac{x}{3}} )
Now, we know that ( 8 = 2^3 ), so we can substitute this into the expression:
[
8^{\frac{x}{3}} = (2^3)^{\frac{x}{3}}
]
Using the power of a power rule again, ( (a^m)^n = a^{m \cdot n} ), we get:
[
(2^3)^{\frac{x}{3}} = 2^{3 \cdot \frac{x}{3}} = 2^x
]
Thus, the expression ( \sqrt[3]{8^x} ) simplifies to ( 2^x ).
Step 3: Matching the answer
Among the options:
- ( \sqrt[x]{8}^x )
- ( 8^{\frac{3}{x}} )
- ( 8^{\frac{x}{3}} )
- ( 8^{3x} )
We can see that the expression ( 8^{\frac{x}{3}} ) directly matches the simplified form of the original expression.
Thus, the correct answer is:
[
8^{\frac{x}{3}}
]
Explanation:
The cube root of ( 8^x ), or ( \sqrt[3]{8^x} ), was simplified using the properties of exponents. The cube root is the same as raising the expression to the ( \frac{1}{3} ) power. We then applied the rule that ( (a^m)^n = a^{m \cdot n} ) to simplify the expression. By substituting ( 8 = 2^3 ), we eventually arrived at the final simplified form of ( 8^{\frac{x}{3}} ), which matches one of the given options.