To find the cube root of \( 0.0216 \) using the factor method, we start by expressing \( 0.0216 \) in a more manageable form.
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Convert to fractions: \[ 0.0216 = \frac{216}{10000} \]
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Factor the numerator and denominator:
- First, factor \( 216 \): \[ 216 = 2^3 \times 3^3 \]
- Next, factor \( 10000 \): \[ 10000 = 10^4 = (2 \times 5)^4 = 2^4 \times 5^4 \]
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Combine the factors: Now, we can express \( \frac{216}{10000} \) in terms of its prime factors: \[ 0.0216 = \frac{2^3 \times 3^3}{2^4 \times 5^4} \] Simplifying this: \[ 0.0216 = \frac{3^3}{2^{4-3} \times 5^4} = \frac{3^3}{2^1 \times 5^4} = \frac{27}{2 \times 625} \]
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Calculate the cube root: The cube root of \( 0.0216 \) can now be found by taking the cube root of the numerator and the denominator separately: \[ \sqrt[3]{0.0216} = \frac{\sqrt[3]{3^3}}{\sqrt[3]{2^1 \times 5^4}} = \frac{3}{\sqrt[3]{2} \times \sqrt[3]{5^4}} \] Since \( \sqrt[3]{5^4} = \sqrt[3]{125} = 5^{4/3} = 5 \times \sqrt[3]{5} \): \[ \sqrt[3]{5^4} = 5 \sqrt[3]{5} \] Thus, \[ \sqrt[3]{0.0216} = \frac{3}{\sqrt[3]{2} \times 5 \sqrt[3]{5}} = \frac{3}{5 \cdot \sqrt[3]{10}} \]
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Exact value: To express \( \frac{3}{5 \cdot \sqrt[3]{10}} \), we can leave it as is, or if we want to get an approximate decimal, we can compute it: \[ \sqrt[3]{10} \approx 2.154 \] Thus, \[ \sqrt[3]{0.0216} \approx \frac{3}{5 \cdot 2.154} \approx \frac{3}{10.77} \approx 0.278. \]
Therefore, the cube root of \( 0.0216 \) is \( 0.278 \) or more precisely expressed as \( \frac{3}{5 \cdot \sqrt[3]{10}} \).
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