Problem 56 - Algebra and Trigonometry

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 solution to \( \sqrt[3]{8} \) is \( 2 \).

Step 1: Understanding the cubic root

The cubic root of a number \( X \) written as \( \sqrt[3]{X} \), is a value that, when cubed (multiplied by itself and then by itself once again), equals \( X \). In this case, we're looking for a number that, when cubed, equals 8.

Step 2: Solving the cubic root

A quick way to identify the cubic root of 8 is to know that \( 2 \times 2 \times 2 = 8 \). Therefore, \( \sqrt[3]{8} = 2 \).

Key Concepts

Cubic Root Evaluation

Cubic root evaluation is the process of finding a number that, when multiplied by itself twice, gives the original number, also known as the radicand. In this instance, evaluating the cubic root of 8 means you're looking for a number that fits the equation:
\[x^3 = 8\]
The solution involves figuring out what value squared, and then multiplied by itself again, equals 8. Since \(2 \times 2 \times 2 = 8\), the cubic root of 8 is 2. This process is integral in algebra as it helps to break down complex equations and solve for unknowns using known relationships.
Understanding cubic roots can aid in identifying the dimensions of geometric shapes, solving physics problems, and more. Unlike square roots, which always pair a number with its inverse to achieve the target value, cubic roots require an odd number of multiplications, making them unique in algebraic operations.

Radicals in Algebra

Radicals are symbols that represent the root of a number. In algebra, they help us manage non-linear relationships and develop solutions for complex equations. The radical symbol \( \sqrt{} \) is most familiar, often representing the square root, but it can also denote other roots, such as cubic roots. The cubic root uses the symbol \( \sqrt[3]{} \) to signify that you need to find a number which, cubed, gives the radicand.

• Radicals help simplify and solve polynomial equations.
• They allow expressions with exponents to be rewritten in more manageable forms.
• Radicals in algebra are used not just for integers, but also for fractions and variables.

Understanding how to manipulate and evaluate radicals is key to progressing in algebraic proficiency. Practices such as isolating radicals and understanding their properties can streamline solving equations involving roots.

Real Numbers

Real numbers are a set of numbers that include all rational and irrational numbers. Anything you can place on a number line is a real number, meaning that they encompass all the numbers we usually work with in algebra and everyday calculations. From
integer values like 3 or -5, to fractions like \(\frac{1}{2}\), and irrational numbers like \(\pi\), they all belong to the real number set.
When dealing with radicals, especially cubic roots, it's important to identify whether the calculated root remains a real number:

• Cubic roots of positive numbers are always real numbers.
• Cubic roots of negative numbers are also real numbers because multiplying three identical negative numbers yields a negative product.
• Whether they are expressed as decimals, fractions, or whole numbers, they can be located on the number line.

Understanding real numbers broadens the capacity to resolve mathematical expressions and substantiate real-world applications.