Which is equivalent to RootIndex 3 StartRoot 8 EndRoot Superscript x?
The Correct Answer and Explanation is :
To find an equivalent expression for (\sqrt[3]{8^x}), we can break it down step by step.
Step 1: Simplify the Cube Root
The expression (\sqrt[3]{8^x}) can be rewritten using the exponent rules. The cube root can be expressed as a fractional exponent:
[
\sqrt[3]{8^x} = (8^x)^{\frac{1}{3}}
]
Step 2: Apply the Power of a Power Rule
Using the power of a power property ((a^m)^n = a^{m \cdot n}), we get:
[
(8^x)^{\frac{1}{3}} = 8^{x \cdot \frac{1}{3}} = 8^{\frac{x}{3}}
]
Step 3: Express 8 in Terms of Powers
Next, we can express 8 as a power of 2. Since (8 = 2^3), we can substitute that into our expression:
[
8^{\frac{x}{3}} = (2^3)^{\frac{x}{3}}
]
Step 4: Simplify Further
Again, using the power of a power property:
[
(2^3)^{\frac{x}{3}} = 2^{3 \cdot \frac{x}{3}} = 2^x
]
Final Result
Thus, the expression (\sqrt[3]{8^x}) simplifies to:
[
\sqrt[3]{8^x} = 2^x
]
Conclusion
In summary, we started with (\sqrt[3]{8^x}) and applied the rules of exponents to simplify it step by step. We used the properties of exponents, including the conversion of the cube root to a fractional exponent, the power of a power rule, and the recognition of 8 as a power of 2. Ultimately, we found that (\sqrt[3]{8^x} = 2^x), making it an equivalent expression that is often simpler and more manageable in further calculations.