Find each root. See Example 3. ∛0.001

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Recognize that the problem asks for the cube root of 0.001, which means finding a number \( x \) such that \( x^3 = 0.001 \).

Express 0.001 as a power of 10: \( 0.001 = 10^{-3} \). This helps simplify the root calculation using exponent rules.

Use the property of roots and exponents: \( \sqrt[3]{10^{-3}} = 10^{\frac{-3}{3}} \). This means you divide the exponent by the root's degree.

Simplify the exponent: \( 10^{\frac{-3}{3}} = 10^{-1} \).

Rewrite \( 10^{-1} \) as a decimal: \( 10^{-1} = 0.1 \), which is the cube root of 0.001.

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Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because 2 × 2 × 2 = 8. It is denoted as ∛x.

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Complex Roots

Decimal Representation and Powers of Ten

Understanding decimals and their relation to powers of ten helps simplify root calculations. For instance, 0.001 can be written as 10⁻³, making it easier to apply root operations using exponent rules.

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Powers Of Complex Numbers In Polar Form (DeMoivre's Theorem)

Properties of Exponents in Root Extraction

Roots can be expressed as fractional exponents, such as ∛x = x^(1/3). Applying this property allows converting roots into exponent form, facilitating calculations especially with powers of ten or other numbers.

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