Problem 24 - Intermediate Algebra

Question

Find each cube root. $$ \sqrt[3]{-125} $$

Step-by-Step Solution

Answer

The cube root of -125 is -5.

Step 1: Understand the Problem

We need to find the cube root of \(-125\), which is the \(x\) such that \(x^3 = -125\).

Step 2: Express the Number as a Power of a Base number

Recognize that \(-125\) can be written as \(-5^3\).

Step 3: Compute the Cube Root Using Properties of Radicals

Using the property of radicals that \(\sqrt[3]{a^3} = a\), we determine\(\sqrt[3]{-5^3} = -5\). Therefore,\(x = -5\).

Step 4: Conclude with the Solution

After confirming that \((-5)^3 = -125\), we conclude that \(-5\) is indeed the correct cube root of \(-125\).

Key Concepts

Understanding Radicals

In mathematics, radicals are used to represent roots of numbers. The most common radical is the square root, but radicals can express other roots, such as cube roots. A cube root is the number that when multiplied by itself twice gives the original number. For instance, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \).
However, cube roots differ from square roots because they can be used with negative numbers. This is because when a negative number is multiplied by itself an odd number of times, the result is negative again. Thus, \( \sqrt[3]{-125} \) is valid and results in a negative number.

• Radicals are a team of symbols representing roots.
• Cube roots search for a number which multiplied thrice returns the original number.
• Unlike square roots, cube roots can include negative input values.

Power with Exponents

Exponents are used to express repeated multiplication of a base number. When dealing with powers, the base is multiplied by itself as many times as indicated by the exponent. For example, \( 5^3 = 5 \times 5 \times 5 = 125 \).

Using exponents can help simplify expressions and solve equations more easily. In the case of cube roots, recognizing a number as a power of another base number (like \(-125 = -5^3\)) helps quickly find the solution.

• Exponents indicate repeated multiplication.
• Recognizing number powers simplifies finding cube roots.
• Negative bases with odd exponents remain negative.

The Realm of Real Numbers

Real numbers are a set of numbers including both rational and irrational numbers. They cover everything from whole numbers, fractions, and decimals — essentially anything that can appear on a number line.

Cube roots, like the one in our problem \( \sqrt[3]{-125} \), fall into the category of real numbers. This is because cube roots of negative numbers (such as \(-5\)) are defined within the real number system. We can visualize this on a number line where both positive and negative values exist.

• Real numbers include all numbers on the number line.
• Cube roots of whole numbers fit within the real number system.
• Even negative numbers and their roots are considered real.

Working with Negative Numbers

Negative numbers appear when subtracting a larger number from a smaller one, or through multiplication involving negative factors. These numbers often confuse students, especially when they occur in complex operations such as cube roots.

In cube roots, negative results are expected when taking the cube root of a negative number. This is important to remember since multiplying a negative number by itself an even number of times makes it positive, but multiplying it an odd number of times (like \(-5 \times -5 \times -5\)) results in a negative number.

• Negative numbers result from subtracting larger values from smaller ones.
• They behave differently based on odd and even power operations.
• Cube roots of negative numbers maintain their negativity.