Find each root. See Example 3.∛0.001

Verified step by step guidance

1. Recognize that finding the cube root of a number is the same as raising that number to the power of \( \frac{1}{3} \).
2. Express the number 0.001 as a power of 10. Since 0.001 is \( 10^{-3} \), rewrite the expression as \( (10^{-3})^{\frac{1}{3}} \).
3. Apply the power of a power property of exponents, which states \( (a^m)^n = a^{m \cdot n} \).
4. Calculate the new exponent by multiplying \(-3\) by \(\frac{1}{3}\), which simplifies to \(-1\).
5. Rewrite the expression as \( 10^{-1} \), which is equivalent to \( \frac{1}{10} \).

Key Concepts

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. It is denoted as ∛x, where x is the number in question. For example, the cube root of 8 is 2, since 2 × 2 × 2 = 8. Understanding cube roots is essential for solving problems involving cubic equations and real-world applications like volume.

Radical Notation

Radical notation is a mathematical notation used to represent roots of numbers. The symbol '√' denotes square roots, while '∛' denotes cube roots. This notation allows for the concise expression of roots, making it easier to perform calculations and understand relationships between numbers. Familiarity with radical notation is crucial for interpreting and solving equations involving roots.

Properties of Exponents

Properties of exponents are rules that govern how to manipulate expressions involving powers and roots. For instance, the property that states a^(1/n) equals the n-th root of a is fundamental when working with roots. These properties help simplify complex expressions and solve equations efficiently, making them vital for understanding higher-level algebra and trigonometry.