Problem 110 - Intermediate Algebra

Question

Choose the correct letter for each exercise. Letters will be used more than once. No pencil is needed. Just think about the meaning of each expression. \(A=2, B=-2, C=\) not a real number $$ 8^{1 / 3} $$

Step-by-Step Solution

Answer

The correct letter for \(8^{1/3}\) is 'A'.

Step 1: Understanding the Expression

We need to determine which letter corresponds to the value of the expression \(8^{1/3}\). The expression \(8^{1/3}\) represents the cube root of 8.

Step 2: Calculate the Cube Root of 8

Evaluate the expression \(8^{1/3}\) by finding the number that, when multiplied by itself twice (3 times total), equals 8. The cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\).

Step 3: Match the Result with Given Options

We found \(8^{1/3} = 2\). Now, according to the options, 2 corresponds to the letter 'A'. Therefore, the correct letter for this expression is 'A'.

Key Concepts

Understanding Cube Roots

Cube roots are a fundamental part of learning about exponents and powers. A cube root of a number is the value that, when multiplied by itself three times, results in the original number. If we denote the cube root of a number as \(x\), the relationship can be expressed as:

• \( x^3 = ext{original number} \)

For example, the cube root of 8 is 2, because when you multiply 2 by itself three times (i.e., \(2 \times 2 \times 2\)), you get 8. This is written mathematically as \( 8^{1/3} = 2 \). Finding cube roots is a common task in exercises involving exponents. Calculating cube roots helps in identifying the base number from which the original number was generated by exponentiation.

Exploring Real Numbers

Real numbers are a broad category within the number system, encompassing all the numbers that can be found on the number line. These include:

• Rational numbers: These can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers, and \(q eq 0\).
• Irrational numbers: Numbers that cannot be expressed as a simple fraction. This includes numbers like \(\pi\) and \(\sqrt{2}\).

The concept of real numbers allows us to perform calculations and solve equations across a wide range of values. In the context of the cube root example, \(8^{1/3} = 2\) is a real number, specifically an integer, which is a type of rational number. Understanding real numbers is crucial for solving a variety of mathematical problems, including those involving square roots, cube roots, and other operations.

Simplifying Mathematical Expressions

Mathematical expressions are combinations of numbers and operators (such as +, -, *, /, etc.) that represent a specific value when calculated. Simplifying these expressions involves reducing them to their simplest form. When simplifying expressions that include exponents:

• Understand the operation indicated—like the cube root denoted by \(^{1/3}\).
• Follow the order of operations to ensure correct calculation. This means performing the calculations in parentheses first, then exponents, followed by multiplication and division, and finally addition and subtraction.

In our cube root example, \(8^{1/3}\) simplifies directly because we recognize \(2^3 = 8\). Simplifying expressions accurately ensures precise outcomes and helps in understanding complex mathematical concepts effectively.