Problem 61 - Algebra 2

Question

Is the product \((1-\sqrt[3]{8})(1+\sqrt[3]{8})\) a rational number? Explain.

Step-by-Step Solution

Answer

Yes, the product \((1-\sqrt[3]{8})(1+\sqrt[3]{8})\) is a rational number, which is \(-3\).

Step 1: Evaluate the Roots

First, we find the cube root of 8 which is 2. Thus, the two expressions become \(1-2\) and \(1+2\). Thereafter, calculate those expressions.

Step 2: Multiply the Expressions

Multiply the two expressions, which results in \((-1)*3\).

Step 3: Evaluate Result

The product evaluates to \(-3\), which is a rational number because it can be expressed as a fraction with an integer in the numerator and a nonzero integer in the denominator. In this case, \(-3\) is equivalent to \(-3/1\).

Key Concepts

Multiplication

Multiplication is a basic arithmetic operation that involves combining numbers or expressions to obtain their product. In our example, we have two expressions:

• The first is \(1 - \sqrt[3]{8}\)
• The second is \(1 + \sqrt[3]{8}\)

To multiply these expressions, you simply follow the distributive law of multiplication, which combines each term of one expression with each term of the other.
In this specific situation, the expressions are represented in a standard form known as the difference of squares, which simplifies calculation. The result of their multiplication reflects the operation \( (a - b)(a + b) = a^2 - b^2 \). Here, the values are:

• \(a = 1\)
• \(b = \sqrt[3]{8} = 2\)

By substituting these values into the formula, the multiplication becomes \(1^2 - 2^2 = 1 - 4 = -3\). The computation leads to \-3\, which is a rational number.

Cube Roots

Understanding cube roots is essential when dealing with problems involving expressions like \(\sqrt[3]{8}\). A cube root of a number is a value which, when multiplied by itself twice, gives the original number. For example, the cube root of 8 is calculated as follows:

• Find a number \(x\) such that \(x \times x \times x = 8\).
• The number \(x\) is 2, because \(2 \times 2 \times 2 = 8\).

To clarify, the cube root \(\sqrt[3]{8}\) is equivalent to a simple integer, here being 2, which simplifies the operation involved in our overall problem.
Cube roots are critical in various mathematical problems as they help in simplifying expressions and determining the pivotal components of polynomial equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. In our problem, the expressions \(1 + \sqrt[3]{8}\) and \(1 - \sqrt[3]{8}\) are examples of simple algebraic expressions.

• Each includes constants: numbers that are added or subtracted, such as 1.
• They also include operations: multiplication and cube roots.

When dealing with algebraic expressions, it’s important to understand the components:

• Terms are the distinct parts of an expression separated by addition or subtraction.
• Factors are components of terms that are multiplied together.

Mastering algebraic expressions enables you to manipulate mathematical models, solve equations, and simplify complex mathematical problems. They serve as building blocks in solving the kind of algebraic problem presented in this exercise. Understanding how to handle them appropriately ensures that you can easily determine results, as seen in turning the original expressions into the result: a rational number.