Problem 17 - Intermediate Algebra

Question

Evaluate each expression. See Example 1. $$ 125^{1 / 3} $$

Step-by-Step Solution

Answer

The value of \( 125^{1/3} \) is 5.

Step 1: Understand the Expression

The expression given is \( 125^{1/3} \). This notation implies taking the cube root of 125. Mathematically, \( 125^{1/3} = \sqrt[3]{125} \).

Step 2: Recall the Definition of Cube Root

The cube root \( \sqrt[3]{x} \) of a number \( x \) is a value \( y \) such that \( y^3 = x \). Our task is to find a number \( y \) such that \( y^3 = 125 \).

Step 3: Identify Perfect Cubes

Recall some small perfect cubes: \( 1^3 = 1 \), \( 2^3 = 8 \), \( 3^3 = 27 \), \( 4^3 = 64 \), \( 5^3 = 125 \). Notice that \( 5^3 = 125 \) matches our number.

Step 4: Conclusion

Since \( 5^3 = 125 \), it follows that \( \sqrt[3]{125} = 5 \). Thus, \( 125^{1/3} = 5 \).

Key Concepts

Perfect Cubes

In mathematics, a perfect cube is a number that can be expressed as the cube of an integer. For example, when we say a number is a perfect cube, it means you can write it in the form of \( n^3 \), where \( n \) is an integer. So, if \( n = 5 \), then \( 5^3 = 125 \). This makes 125 a perfect cube.
Recognizing perfect cubes can be very useful in solving problems involving cube roots. It's good to memorize a few small perfect cubes for quick reference:

• \( 1^3 = 1 \)
• \( 2^3 = 8 \)
• \( 3^3 = 27 \)
• \( 4^3 = 64 \)
• \( 5^3 = 125 \)

Knowing these values off the top of your head helps quickly identify when you are dealing with perfect cubes. When you see a number like 125, you immediately recognize it as \( 5^3 \). This is the key to calculating cube roots swiftly.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. In the expression \( a^b \), \( a \) is the base and \( b \) is the exponent. This operation tells you to multiply the base by itself \( b \) times.
For cube roots, the exponent is \( \frac{1}{3} \), meaning you are trying to find which number, when cubed, equals the original number. Conversely, raising a number to the power of 3 (cubing a number) means multiplying the base three times by itself. For instance, \( 5^3 = 5 \times 5 \times 5 = 125 \).
When dealing with cube roots, the notation \( n^{1/3} \) signifies the cube root of \( n \). This is because exponentiation with a fractional exponent is equivalent to a root operation. Thus, \( 125^{1/3} = \sqrt[3]{125} \), pinpointing the original integer that, when raised to the power of 3, produces the number.

Evaluating Expressions

Evaluating expressions involves simplifying or solving them to find an equivalent number or answer. Let's break down how to evaluate \( 125^{1/3} \):

• Step 1: Translate the fraction exponent into a root operation, where \( 125^{1/3} = \sqrt[3]{125} \).
• Step 2: Recall the concept of finding perfect cubes. You need to determine which number, cubed, equals 125. From earlier, we know \( 5^3 = 125 \).
• Step 3: Since we found that \( y = 5 \) satisfies \( y^3 = 125 \), we conclude that \( \sqrt[3]{125} = 5 \).

It helps to verify calculations when tackling similar expressions. Evaluating expressions efficiently requires practice with converting between exponential and root forms, as well as confidence with perfect cubes. As you practice, these concepts will become second nature!