Problem 56 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(55-66,\) or indicate that the root is not a real number. $$\sqrt[3]{8}$$

Step-by-Step Solution

Answer

The cube root of 8 is 2.

Step 1: Understand the Cube Root

The cube root of a number is a value that, when multiplied by itself twice, gives the original number. It is denoted as \(\sqrt[3]{n}\), where `n` is the number for which we want to find the cube root.

Step 2: Compute the Cube Root

To compute the cube root of 8, we ask: which number times itself twice equals 8? The answer is 2, since \(2*2*2 = 8\). Therefore, the cube root of 8 is 2.

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics, encompassing all the numbers that can be found on the number line. This includes both rational and irrational numbers.

• Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, such as 1/2 or 4.
• Irrational numbers, on the other hand, cannot be expressed as a simple fraction, like √2 or π.

Real numbers play a crucial role in many areas of math and real-world applications. They include positive numbers, negative numbers, and zero, providing a complete system for solving equations and evaluating expressions. It’s important to know that cube roots can be asked for any real number, though not every cube root results in a real number.

Root Evaluation

Root evaluation refers to the process of finding root values of numbers, typically square roots or cube roots. Evaluating roots helps us determine which number, when repeatedly multiplied, gives the original number.Cube roots are written as \(\sqrt[3]{n}\), representing the value when multiplied by itself two more times equals n.To find the cube root of 8 as an example, we look for a number which when raised to the power of three (\[ x^3 = 8 \]) equals 8.The number 2 satisfies this equation because:

• 2 x 2 = 4
• 4 x 2 = 8

Therefore, the cube root of 8 is 2.When evaluating cube roots of different numbers, it's useful to remember these steps and explore potential integer solutions.This understanding aids in quickly solving root-related math problems.

Mathematical Expressions

Mathematical expressions are a way to represent numbers, operations, and relationships, often used to solve problems and convey mathematical instructions. These can include numbers, variables, operations (like addition, subtraction), and exponents. They allow us to work with equations and perform calculations in a structured manner.For example, the expression \(\sqrt[3]{8}\) is a cube root expression, where 8 is the radicand (the number under the radical sign).This expression tells us to find a number that, when used in the operation of cubing (multiplying by itself twice), results in 8.

• Recognizing and correctly understanding these expressions is vital for problem-solving.
• Such expressions are used widely in algebra, calculus, and beyond.
• Knowing how to interpret them helps in translating real-world problems into mathematical ones, making solutions achievable.

Understanding expressions thus forms a core skill in mathematical reasoning and application.