Problem 73 - 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: Understand the Cube Root

The expression \( \sqrt[3]{-125} \) represents the cube root of \(-125\). The cube root of a number \(-x\) is a value \(y\) such that \(y^3 = -x\).

Step 2: Find the Cube Root of Positive 125

Calculate the cube root of \(125\) first, ignoring the negative sign. We need to determine the number that when multiplied by itself twice more equals \(125\). The cube root of \(125\) is \(5\) because \(5^3 = 5 \times 5 \times 5 = 125\).

Step 3: Apply the Negative Sign

Since we originally had \(\sqrt[3]{-125}\), we need to apply the sign after simplifying the cube root of the absolute value. So, \(\sqrt[3]{-125} = -5\) because \((-5)^3 = -125\).

Key Concepts

Negative Numbers

Negative numbers are numbers with a value less than zero. They are typically represented with a minus sign in front of them. Understanding how negative numbers interact with basic math operations is essential, especially when dealing with cube roots.

For instance, when you take the cube root of a negative number, you're looking for a number that, when cubed, will give you the negative value you started with. This is slightly different from taking the square root of a negative number, which results in an imaginary number. With cube roots, however, you can still find a real number solution because tripling a negative number yields another negative number.

Here's an easy way to think about it:

• If the original number is negative, its cube root will also be negative.
• If the original number is positive, its cube root will also be positive.

Understanding these properties makes it easier to deal with cube roots of negative numbers.

Exponentiation

Exponentiation is the process of raising a number to a power. In simple terms, it means multiplying a number by itself a certain number of times. For cube roots, we are particularly interested in the power of three.

When you're asked to find the cube root of a number, you're looking for a value that, when raised to the third power (cubed), gives the original number back.

In mathematical terms, if you have a number \( x \), the cube root is the number \( y \) such that \( y^3 = x \). This is essentially the reverse of cubing, which is an example of exponentiation.

• The expression \(5^3 = 125\) shows how 5, when used as a factor three times, results in 125.
• To find \( \sqrt[3]{-125} \), you recognize that \((-5)^3 = -125\). This tells you that -5, when cubed, equals -125.

Understanding exponentiation is vital when simplifying cube roots, ensuring you can work backward from a cubed value.

Simplification Process

Simplification makes a math problem more manageable by reducing it to its simplest form, often involving steps like breaking down numbers or using specialized knowledge like cube roots.

When simplifying the cube root of a number, especially a negative one like \( -125 \), there are distinct stages to follow:

• First, take the absolute value of the number to find the cube root as if it were positive. Here, you deal with \( 125 \).
• Next, calculate which number cubed equals this positive value. \( 5 \times 5 \times 5 = 125 \), so the cube root of \( 125 \) is \( 5 \).
• Finally, reapply the original negative sign to this result. Thus, \( \sqrt[3]{-125} = -5 \).

Following these steps ensures the cube root is correctly simplified, making it easier to interpret the outcome and relate it to further mathematical tasks.