Problem 73 - Elementary and Intermediate Algebra

Question

Simplify each cube root. See Example \(6 .\) $$ \sqrt[3]{-125} $$

Step-by-Step Solution

Answer

The cube root of \(-125\) is \(-5\).

Step 1: Understanding the Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In the expression \( \sqrt[3]{-125} \), we are looking for a number that, cubed (raised to the power of 3), results in \(-125\).

Step 2: Determine the Cube Root of Positive Equivalent

First, find the cube root of the positive equivalent number. The number 125 is a perfect cube, as it can be expressed as \(5^3\). Since \(5 \times 5 \times 5 = 125\), the cube root of 125 is 5.

Step 3: Adjust for the Negative Sign

Since cube roots can be taken for negative numbers, and we know the cube root of 125 is 5, the cube root of \(-125\) is \(-5\). This is because \(-5 \times -5 \times -5 = -125\).

Step 4: Conclusion

Hence, the simplified cube root of \(\sqrt[3]{-125} \) is \(-5\).

Key Concepts

Simplifying Expressions

Simplifying expressions can often seem tricky, but it becomes easier with practice. To simplify means to reduce an expression into its simplest form, which often involves breaking it down into components that are easier to manage or understand. It can involve arithmetic operations, breaking down complex numbers, or identifying root values.
For cube roots, simplifying starts with identifying whether the number is a perfect cube. This means it can be expressed as the cube of an integer. Once the perfect cube is identified, the expression can be simplified considerably just like in our example with \( \sqrt[3]{-125} \).
Simplifying expressions allows calculations to be handled more efficiently and is a fundamental aspect of solving mathematical problems. It helps in recognizing patterns and using them to perform more complex tasks. It's useful to practice simplifying expressions regularly.

Negative Numbers in Roots

When working with cube roots, it is important to understand how negative numbers behave. Unlike square roots, cube roots can handle negative numbers quite comfortably. This is because multiplying three negative numbers results in a negative outcome.
In simpler terms:

• A negative number multiplied by itself twice becomes positive.
• Multiply it one more time by a negative, and it turns negative again, making it possible to find the cube root of a negative number.

This is evident with \( \sqrt[3]{-125} \). The cube root turns out to be \(-5\), as \(-5 \times -5 \times -5 = -125\). Embracing the unique properties of cube roots with negative numbers can open doors to solving various algebraic problems.

Perfect Cubes

Perfect cubes are numbers that can be expressed as the cube of an integer. Recognizing these numbers can significantly simplify operations involving cube roots. To determine if a number is a perfect cube:

• Identify an integer that, when multiplied by itself twice, gives the number in question.
• For instance, \(125 \) is a perfect cube since \(5^3 = 125 \).

Recognizing perfect cubes helps not only in simplifying expressions but also in swiftly identifying cube roots.
In mathematical problems, spotting these cubes can lead to quicker solutions and less computational complexity. By practicing identifying perfect cubes, students develop the ability to simplify and solve cube root problems successfully.