Problem 56 - College Algebra

Question

Evaluate each expression or indicate that the root is not a real number. $$\sqrt[3]{8}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Cube Root Symbol

The notation \(\sqrt[3]{8}\) stands for the cube root of 8. The number 3 in the root symbol denotes that the operation to be performed is the cube (third) root.

Step 2: Evaluating the Cube Root

The cube root of 8 can be determined by asking: 'Which number multiplies by itself three times to give 8?'. The answer is 2, as \(2 × 2 × 2 = 8\).

Step 3: Final Answer

Therefore, the cube root of 8 is 2.

Key Concepts

Radical Expressions

Radical expressions are mathematical terms that involve roots, such as square roots, cube roots, or any higher roots. These expressions use radical symbols to indicate root operations.
The general symbol for a root is \(\sqrt[n]{x}\), where \(n\) is the degree of the root and \(x\) is the number being rooted.

• The most common radical expression is the square root, \(\sqrt{x}\), where \(n = 2\) and is often written without the 2.
• For a cube root, like \(\sqrt[3]{x}\), \(n = 3\).
• Radical expressions allow us to handle non-integer exponents.

By understanding radical expressions, one can simplify and evaluate them, even for larger numbers.

Real Numbers

Real numbers include all the numbers we commonly use and encounter in everyday life. They encompass both rational and irrational numbers.
Rational numbers are numbers that can be expressed as the fraction of two integers, such as \(\frac{1}{2}\) or -3.

• Integers, whole numbers, and fractions are examples of rational numbers.
• Irrational numbers, like \(\pi\) or \(\sqrt{2}\), cannot be expressed as simple fractions.

A root is a real number if it can be simplified to a number on the number line, like 2 from \(\sqrt[3]{8}\).
This helps determine if a radical expression produces a real value.

Evaluating Expressions

Evaluating expressions is the process of simplifying or solving mathematical statements to find their value.

• To evaluate a cube root, ask: Which number multiplied by itself three times equals the original number?
• This question helps break down the radical expression to its simplest form.
• For instance, in \(\sqrt[3]{8}\), ask: What multiplied by itself thrice equals 8? The answer is 2.

Evaluating ensures you reach the simplest and most specific answer possible.
It transforms complex expressions into understandable numbers.