3:25
Simplifying Radical Expressions 1
Khan Academy
Overview
This video explains how to simplify radical expressions, specifically focusing on finding the cube root of a negative number. It demonstrates that finding the cube root of a negative number is equivalent to finding a number that, when multiplied by itself three times, results in that negative number. The process involves prime factorization, separating the negative component, and identifying the cube root of both the negative one and the remaining positive factor.
How was this?
Save this permanently with flashcards, quizzes, and AI chat
Chapters
- The cube root of a negative number can be expressed as that number raised to the power of 1/3.
- Finding the cube root of a negative number means finding a value that, when cubed (multiplied by itself three times), equals the original negative number.
- This is represented mathematically as: if x³ = -343, then x = ³√(-343).
Understanding this concept is crucial for solving equations and simplifying expressions involving odd roots of negative numbers, which appear in various mathematical and scientific contexts.
If x³ = -343, then x is the cube root of -343.
- To find the cube root of -343, begin by factoring out -1, resulting in -1 * 343.
- Next, perform prime factorization on the positive number (343). Test divisibility by prime numbers starting with 2, 3, 5, and then 7.
- 343 is divisible by 7, yielding 49. Since 49 is 7 * 7, the prime factorization of 343 is 7 * 7 * 7.
Prime factorization breaks down a complex number into its fundamental building blocks, making it easier to identify perfect cubes and simplify radical expressions.
The prime factorization of 343 is 7 x 7 x 7.
- Rewrite the original expression using the prime factorization: ³√(-343) = ³√(-1 * 7 * 7 * 7).
- Separate the cube root of the negative one and the cube root of the positive factors: ³√(-1) * ³√(7 * 7 * 7).
- The cube root of -1 is -1, because (-1) * (-1) * (-1) = -1.
- The cube root of (7 * 7 * 7) is 7, because 7 * 7 * 7 = 343.
- Multiply the results: -1 * 7 = -7.
This step-by-step calculation demonstrates how to systematically solve for the cube root by leveraging the properties of radicals and prime factorization, leading to the final simplified answer.
The cube root of (-1 * 7 * 7 * 7) is calculated as the cube root of -1 (which is -1) multiplied by the cube root of (7 * 7 * 7) (which is 7), resulting in -7.
Key takeaways
- The cube root of a negative number is always a negative number.
- Prime factorization is a key strategy for simplifying radical expressions.
- The cube root of a number cubed is the number itself (e.g., ³√(7³) = 7).
- The cube root of -1 is -1.
- To find the cube root of a negative number, factor out -1 and find the cube root of the remaining positive number, then multiply by -1.
- Verifying the answer by cubing the result confirms its correctness.
Key terms
Cube rootRadical expressionExponentPrime factorizationDivisibilityNegative onePositive factor
Test your understanding
- What does it mean to find the cube root of a negative number?
- How can prime factorization help in simplifying cube roots of negative numbers?
- What is the cube root of -1, and why?
- Explain the process of finding the cube root of -343 using the steps shown in the video.
- How can you verify that -7 is the correct cube root of -343?