Pregunta
Which expression is equivalent to \( 24^{\frac{1}{3}} \) ? \( 2 \sqrt{3} \) \( 2 \sqrt[3]{3} \) \( 2 \sqrt{6} \) \( 2 \sqrt[3]{6} \)
Ask by Norton Salazar. in the United States
Feb 03,2025
Solución de tutoría real
Respuesta verificada por el tutor
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The equivalent expression is \( 2 \sqrt[3]{3} \).
Solución
To determine which expression is equivalent to \( 24^{\frac{1}{3}} \), let's simplify \( 24^{\frac{1}{3}} \):
1. **Factorize 24:**
\[
24 = 8 \times 3
\]
2. **Apply the cube root:**
\[
24^{\frac{1}{3}} = (8 \times 3)^{\frac{1}{3}} = 8^{\frac{1}{3}} \times 3^{\frac{1}{3}}
\]
3. **Simplify each part:**
\[
8^{\frac{1}{3}} = 2 \quad \text{(since \(2^3 = 8\))}
\]
\[
3^{\frac{1}{3}} = \sqrt[3]{3}
\]
4. **Combine the results:**
\[
24^{\frac{1}{3}} = 2 \times \sqrt[3]{3}
\]
Therefore, the equivalent expression is:
**\( 2 \sqrt[3]{3} \)**
**Answer:** \( 2 \sqrt[3]{3} \)
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución The Deep Dive
To find an expression equivalent to \( 24^{\frac{1}{3}} \), let's break it down. We can rewrite 24 as \( 2^3 \times 3 \). Therefore: \[ 24^{\frac{1}{3}} = (2^3 \times 3)^{\frac{1}{3}} = 2^{3 \cdot \frac{1}{3}} \times 3^{\frac{1}{3}} = 2 \times \sqrt[3]{3} \] So, the correct choice is \( 2 \sqrt[3]{3} \). Also, it's helpful to know that working with exponents and roots can often be simplified by breaking numbers down into their prime factors, which allows for easier manipulation and understanding of their relationships!
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