Question
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Answer
1. \( \sqrt{72} = 6\sqrt{2} \)
2. \( \sqrt[3]{32} = 2\sqrt[3]{4} \)
Solution
1. Empezamos con \( \sqrt{72} \):
- Factorizamos el radicando en factores cuadrados:
\[
72 = 36 \times 2
\]
- Aplicamos la propiedad de la raíz:
\[
\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}
\]
2. Ahora, simplificamos \( \sqrt[3]{32} \):
- Escribimos \( 32 \) como potencia de \( 2 \):
\[
32 = 2^5
\]
- Separamos la potencia en un factor que sea múltiplo de \( 3 \):
\[
2^5 = 2^3 \cdot 2^2
\]
- Aplicamos la propiedad de la raíz cúbica:
\[
\sqrt[3]{32} = \sqrt[3]{2^3 \cdot 2^2} = \sqrt[3]{2^3} \cdot \sqrt[3]{2^2} = 2 \sqrt[3]{4}
\]
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Simplify this solution Extra Insights
Did you know that \( \sqrt{72} \) can be broken down into \( \sqrt{36 \times 2} \)? This simplifies beautifully to \( 6\sqrt{2} \). It’s a fun twist on working with squares since 6 is the square root of 36, making your math life a little easier! For \( \sqrt[3]{32} \), we find that 32 is actually \( 2^5 \), or \( 2 \) multiplied by itself five times. Therefore, this cube root simplifies to \( 2^{5/3} \) or \( 2^{1 + 2/3} = 2 \cdot \sqrt[3]{4} \), which balances between nice neat numbers and roots—perfect for impressing friends at your next study session!
