Question
\begin{tabular}{l} 2. Evaluate: \( \sqrt[3]{8^{2}}= \) \\ \hline Simplify the surd: \( \sqrt{200} \) \\ Answer: \\ \hline Evaluate: \( \sqrt[8]{63^{8}} \) \\ Answer: \\ \hline\end{tabular}
Ask by Moreno Frank. in Papua New Guinea
Mar 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
1. \( \sqrt[3]{8^{2}} = 4 \)
2. \( \sqrt{200} = 10\sqrt{2} \)
3. \( \sqrt[8]{63^{8}} = 63 \)
Solution
1. Evaluate \( \sqrt[3]{8^{2}} \)
- Express \(8\) as a power of 2: \(8 = 2^3\).
- Then, \(8^2 = (2^3)^2 = 2^{6}\).
- Now, take the cube root:
\[
\sqrt[3]{2^{6}} = 2^{\frac{6}{3}} = 2^{2} = 4.
\]
2. Simplify the surd \( \sqrt{200} \)
- Factorize 200 as \(200 = 100 \times 2\).
- Then,
\[
\sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \cdot \sqrt{2} = 10\sqrt{2}.
\]
3. Evaluate \( \sqrt[8]{63^{8}} \)
- Recognize that \(\sqrt[8]{63^{8}}\) represents the eighth root of \(63^{8}\), which gives back the original number (since \(63\) is positive):
\[
\sqrt[8]{63^{8}} = 63.
\]
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Simplify this solution Mind Expander
First additional information: To tackle \( \sqrt[3]{8^{2}} \), remember that \( 8 \) can be expressed as \( 2^3 \). Therefore, \( 8^{2} = (2^3)^{2} = 2^{6} \). Now, taking the cube root gives us \( \sqrt[3]{2^{6}} = 2^{2} = 4 \). Second additional information: When simplifying \( \sqrt{200} \), it’s helpful to prime factor the number. \( 200 = 100 \times 2 = 10^2 \times 2 = (10\sqrt{2})^2 \). So, \( \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \). For \( \sqrt[8]{63^{8}} \), simply recognize that the eighth root and the exponent cancel each other out, giving you \( 63 \).
