Problem 111 - 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 answer is B.

Step 1: Understand the Expression

We are given the expression \(-8^{1/3}\) which denotes the cube root of -8. We need to determine if this expression evaluates to 2, -2, or is not a real number.

Step 2: Evaluating the Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In this case, we need a number \(x\) such that \(x^3 = -8\).

Step 3: Calculate the Cube Root

The number \(-2\) when cubed gives \((-2)^3 = -8\). Therefore, the cube root of \(-8\) is \(-2\).

Step 4: Choose the Correct Letter

Since the expression evaluates to \(-2\), we assign the letter "B" to this expression.

Key Concepts

Real Numbers

Real numbers are a fundamental part of mathematics encompassing all the numbers you encounter in daily life. These include:

• Natural numbers (like 1, 2, 3, etc.)
• Whole numbers (which include natural numbers and zero)
• Integers (which include positive and negative whole numbers)
• Rational numbers (like fractions and decimals that can be expressed as a quotient of integers)
• Irrational numbers (such as pi or the square root of any non-perfect square) that cannot be expressed as simple fractions

Real numbers are everywhere and can represent continuous values without gaps. They are used in measuring distances, lengths, temperatures, and just about any real-world quantity. When dealing with expressions like \(-8^{1/3}\), understanding that the result is a real number helps solidify the expression within the realm of standard math.
The expression involves a cube root, which is a part of the real numbers because \(-2\) is a valid integer that satisfies the cube root of \(-8\). Thus, even negative and non-integer values find their place on the continuum of real numbers.

Evaluating Expressions

Evaluating expressions means finding out what an expression is equal to when all the variables are replaced with their actual values. This often involves arithmetic operations like addition, subtraction, multiplication, and division, as well as working with roots and exponents.
In the expression \(-8^{1/3}\), evaluating it means calculating the cube root of \(-8\). A cube root is a specific case where you find a number that multiplies by itself three times to reach the given value. Here, we determine that \(-2\) is the number that fulfills this requirement, as it satisfies the equation \((-2)^3 = -8\).
When evaluating, you may encounter the need to simplify the expression by handling each part of it step by step. This ensures that the final outcome is as exact as possible, reflecting the true value of the original expression.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operation symbols (+, -, *, /) used to represent a particular set of values. Unlike equations, they do not have equal signs, which means they show relationships rather than statements of equality.
Expressions like \-8^{1/3}\ require comprehension of operations and exponents. Here, the focus is on the cube root part, which indicates a repeated multiplication. Algebraic manipulation can involve substituting numbers for variables or combining similar terms.
In this case, when we see expressions involving cube roots, it’s essential to know that the operation denotes finding one of the three equal factors of the number involved. For example, finding the cube root of \(-8\) can be interpreted in terms of algebra as an expression that outputs \(-2\), contextualized within the rules governing real numbers and operations on them.