Express $8^{\frac{2}{3}}$ in simplest radical form.
Understand the Problem
The question asks us to simplify the expression $8^{\frac{2}{3}}$ into its simplest radical form. This involves understanding fractional exponents and how they relate to radicals, as well as simplifying radicals where possible.
Answer
4
Answer for screen readers
4
Steps to Solve
- Express the fractional exponent as a radical
Recall that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. Therefore, $8^{\frac{2}{3}}$ can be expressed as $\sqrt[3]{8^2}$.
- Simplify inside the radical
Calculate $8^2$, which equals 64. So we now have $\sqrt[3]{64}$.
- Evaluate the cube root
Find the cube root of 64. Since $4^3 = 4 \cdot 4 \cdot 4 = 64$, the cube root of 64 is 4. Therefore, $\sqrt[3]{64} = 4$.
4
More Information
The expression $8^{\frac{2}{3}}$ simplifies to the integer 4.
Tips
A common mistake is to take the square root instead of the cube root, or miscalculating $8^2$. Also, some students might try to take the cube root of 8 first, and then square the result. While this is correct, it might cause issues with more complex problems
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