Find each root. See Example 3.-∛512

Verified step by step guidance

1. Identify the expression: We need to find the cube root of -512, which is written as \(-\sqrt[3]{512}\).
2. Understand the cube root: The cube root of a number \(x\) is a number \(y\) such that \(y^3 = x\).
3. Consider the negative sign: Since we are dealing with a negative number, the cube root will also be negative because the cube of a negative number is negative.
4. Break down 512: Recognize that 512 can be expressed as \(2^9\) because \(2^9 = 512\).
5. Calculate the cube root: Since \(512 = 2^9\), the cube root of 512 is \(2^{9/3} = 2^3 = 8\). Therefore, the cube root of -512 is \(-8\).

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. For example, the cube root of 512 is the number that satisfies the equation x³ = 512.

Radical Notation

Radical notation is a way to express roots of numbers using the radical symbol (√). For cube roots, the notation is ∛, indicating the root is taken to the third degree. Understanding this notation is essential for solving problems involving roots.

Prime Factorization

Prime factorization involves breaking down a number into its prime factors, which can simplify the process of finding roots. For instance, 512 can be expressed as 2^9, making it easier to determine its cube root by dividing the exponent by 3.