Question
Which of the following is equal to the expression below \( (8 \cdot 320)^{\frac{1}{3}} \) A. 40 B. \( 8 \sqrt[3]{5} \) C. 30 D. \( 10 \sqrt[3]{5} \)
Ask by George Herrera. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The answer is \( 8 \sqrt[3]{5} \), choice B.
Solution
To determine which option is equal to the expression \( (8 \cdot 320)^{\frac{1}{3}} \), let's simplify the expression step by step.
1. **Multiply the values inside the parentheses:**
\[
8 \cdot 320 = 2560
\]
2. **Express 2560 in terms of its prime factors:**
\[
2560 = 2^9 \cdot 5
\]
3. **Apply the cube root to the prime factorization:**
\[
(2^9 \cdot 5)^{\frac{1}{3}} = 2^{3} \cdot 5^{\frac{1}{3}} = 8 \cdot \sqrt[3]{5}
\]
So, the simplified form of \( (8 \cdot 320)^{\frac{1}{3}} \) is \( 8 \sqrt[3]{5} \).
**Answer:**
**B. \( 8 \sqrt[3]{5} \)**
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Simplify this solution Bonus Knowledge
To simplify \( (8 \cdot 320)^{\frac{1}{3}} \), let's first calculate \( 8 \cdot 320 \). We have: \[ 8 \cdot 320 = 2560 \] Now, we need to find \( 2560^{\frac{1}{3}} \). It can be beneficial to factor \( 2560 \): \[ 2560 = 8 \cdot 320 = 2^3 \cdot 2^6 \cdot 5 = 2^{9} \cdot 5 \] Next, we take the cube root: \[ (2^{9} \cdot 5)^{\frac{1}{3}} = 2^{9/3} \cdot 5^{1/3} = 2^3 \cdot 5^{1/3} = 8 \cdot \sqrt[3]{5} \] So \( (8 \cdot 320)^{\frac{1}{3}} = 8 \sqrt[3]{5} \). Therefore, the correct answer is: B. \( 8 \sqrt[3]{5} \)
