Problem 78 - Algebra for College Students

Question

Find the indicated products. Assume all variables that appear as exponents represent positive integers. $$\left(x^{a}+4\right)\left(x^{a}-9\right)$$

Step-by-Step Solution

Answer

The product is \(x^{2a} - 81\).

Step 1: Identify the expression types

The expression \((x^{a} + 4)(x^{a} - 9)\) is a product of two binomials of the form \((u + v)(u - w)\), where \(u = x^a\), \(v = 4\), and \(w = 9\). This product can be simplified using the difference of squares formula.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula states that \((u + v)(u - w) = u^2 - w^2\). In our case, apply this formula with \(u = x^a\), \(v = 4\), and \(w = 9\). So the expression becomes \((x^a)^2 - 9^2\).

Step 3: Simplify the Expression

Calculate each square: \((x^a)^2 = x^{2a}\) and \(9^2 = 81\). Therefore, the expression simplifies to \(x^{2a} - 81\).

Key Concepts

Binomials

Understanding binomials is crucial in algebra as they are core components of many algebraic expressions. A binomial is an algebraic expression that contains exactly two terms. These terms are often separated by a plus "+" or minus "-" sign. In the given exercise, the binomials are \(x^{a} + 4\) and \(x^{a} - 9\). Each of these expressions features a variable term \(x^{a}\) and a constant term, 4 in the first binomial and -9 in the second.

When you encounter binomials, you can use specific algebraic formulas to simplify their products or expansions. One common formula used is the difference of squares, which applies when you have two binomials in a specific form, \(u + v\) and \(u - w\). This formula allows certain expressions to be simplified quickly and easily.

Exponents

Exponents are a powerful way to represent repeated multiplication. In the exercise, we have the variable \(x\) raised to the power \(a\), or \(x^{a}\). The exponent \(a\) indicates how many times \(x\) is multiplied by itself.
For example, if \(a = 3\), then \(x^{a} = x \cdot x \cdot x\).

In algebra, working with exponents involves understanding the rules of operations such as the product, quotient, and power rules. Particularly, in the difference of squares formula, you encounter exponents when squaring terms. In our solution, we see \((x^{a})^2\) which simplifies through the power rule to \(x^{2a}\). Exponents can significantly change the value of a term, thus understanding them is vital for simplifying and solving algebraic expressions.

Algebraic Expressions

Algebraic expressions consist of variables, coefficients, and constants combined using mathematical operations. They represent a range of numbers depending on the values of the variables involved. In the exercise, the algebraic expression to be simplified is \(\left(x^{a}+4\right)\left(x^{a}-9\right)\).

Algebraic expressions can be manipulated using certain algebraic identities or formulas to simplify them further. The difference of squares is one such formula that specifically helps in dealing with the product of binomials. By applying such formulas, complex expressions can be reduced to simpler terms. This can significantly ease the problem-solving process.

It is also essential to correctly follow the order of operations (PEMDAS/BODMAS) to ensure accurate results. Learning to navigate through different expressions and simplifying them effectively is a critical skill in algebra.