Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

0. Review of College Algebra

Rationalizing Denominators

Trigonometry

0. Review of College Algebra

Rationalizing Denominators

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1:22 minutes

Problem 50

Textbook Question

Find each root. See Example 3.∛0.125

Verified step by step guidance

Recognize that finding the cube root of a number is the same as raising that number to the power of \( \frac{1}{3} \).

Express the number 0.125 as a fraction: \( 0.125 = \frac{1}{8} \).

Recall that \( 8 = 2^3 \), so \( \frac{1}{8} = (2^3)^{-1} = 2^{-3} \).

Apply the property of exponents: \( (2^{-3})^{\frac{1}{3}} = 2^{-3 \times \frac{1}{3}} = 2^{-1} \).

Conclude that the cube root of 0.125 is \( 2^{-1} \), which is the same as \( \frac{1}{2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 0.125 is the number that satisfies the equation x³ = 0.125. This concept is essential for solving problems involving cubic equations and understanding the relationship between exponents and roots.

Radical Notation

Radical notation is a mathematical notation used to represent roots of numbers. The cube root is denoted by the radical symbol with a small '3' placed outside, such as ∛x. Understanding this notation is crucial for interpreting and solving problems that involve roots, as it provides a clear way to express the operation being performed.

Decimal Representation

Decimal representation refers to expressing numbers in the base-10 system, which is commonly used in mathematics. In the context of cube roots, converting fractions like 0.125 into their decimal form helps in visualizing and calculating roots. Recognizing how to manipulate and interpret decimal numbers is vital for accurate computations in trigonometry and algebra.