Problem 55 - College Algebra

Question

Evaluate each expression or indicate that the root is not a real number. $$\sqrt[3]{125}$$

Step-by-Step Solution

Answer

The cube root of 125 is 5.

Step 1: Understanding the Problem

Here it’s necessary to find the cube root of a given number. In other words, find a number \( x \) such that \( x \times x \times x = 125 \). It's crucial to know that the cube root is the reverse operation of cubing a number.

Step 2: Compute the Cube Root

The cube root of a number \( b \) is the number \( a \) such that \( a^3 = b \). Here \( b = 125 \). Looking at the cube numbers we know, we can see that \( 5^3 = 125 \). So, the cube root of 125 is 5.

Key Concepts

Radicals in Algebra

When dealing with radicals in algebra, it's important to understand they represent the opposite operation of exponentiation. The most common radical you'll encounter is the square root, but radicals can also represent other roots, such as the cube root, fourth root, and so on. A radical expression can be written in the form \( \sqrt[n]{a} \) where \( a \) is called the radicand and \( n \) is the index indicating the degree of the root.

For example, in the cube root expression \( \sqrt[3]{125} \) from our exercise, 125 is the radicand, and the index 3 indicates we are finding a number that when multiplied by itself three times gives us 125. To simplify a radical, we look for a number or expression that, when raised to the power of the index, matches the radicand. The properties of radicals are similar to those of exponents, which means they obey rules for multiplication, division, and exponentiation. Simplifying radicals enhances understanding and solving complex algebraic equations.

Cube Numbers

Understanding cube numbers is essential when working with cube roots. A cube number, or a perfect cube, is a number that is the cube of an integer. This means if you take an integer \( n \) and multiply it by itself twice, you get a cube number: \( n^3 \). So, if we cube the number 2, we get \( 2^3 = 2 \times 2 \times 2 = 8 \), making 8 a cube number.

Knowing the list of cube numbers (\( 1^3, 2^3, 3^3, 4^3 \) and so on) can significantly simplify the process of finding cube roots without the need for calculators or lengthy computations. Recognizing that 125 is a cube number because \( 5^3 = 125 \) allows you to immediately identify its cube root as 5. Familiarity with cube numbers can also help in identifying when a number does not have an integer as its cube root, leading to an irrational or non-real number.

Exponents and Roots

The relationship between exponents and roots is fundamental in algebra. Exponents represent repeated multiplication, where \( a^n \) signifies a number \( a \) multiplied by itself \( n \) times. Conversely, roots are about identifying a number that, when raised to a certain power, gives the original number.

For example, the cube root is directly related to the concept of cubing a number. The cube root of \( b \) (written as \( \sqrt[3]{b} \) or \( b^{\frac{1}{3}} \) in exponential form) finds the number \( a \) that satisfies the equation \( a^3 = b \). If we reverse the operation and cube \( a \) (i.e. \( a^3 \)), we get \( b \) again. This inverse relationship helps us to understand and compute roots without memorization but through conceptual understanding. Learning these concepts provides a strong foundation for more complex algebraic operations involving irrational numbers, polynomial equations, and functions.