Problem 82 - Algebra for College Students

Question

Find the indicated products. Assume all variables that appear as exponents represent positive integers. $$\left(x^{2 a}+6\right)\left(x^{2 a}-4\right)$$

Step-by-Step Solution

Answer

The product is \(x^{4a} - 16\).

Step 1: Identify the Structure of the Expression

The given expression is a product of two binomials: \((x^{2a} + 6)\) and \((x^{2a} - 4)\). This can be recognized as a difference of squares formula, which is \((a + b)(a - b) = a^2 - b^2\). Here, \(a = x^{2a}\) and \(b = 4\).

Step 2: Apply the Difference of Squares Formula

Apply the difference of squares formula:\[(a + b)(a - b) = a^2 - b^2\]Substitute \(a = x^{2a}\) and \(b = 4\) into the formula:\[(x^{2a} + 6)(x^{2a} - 4) = (x^{2a})^2 - 4^2\]

Step 3: Simplify Each Term

Now, simplify each term separately:- The term \((x^{2a})^2\) simplifies to \(x^{4a}\) because the exponent rule \((x^m)^n = x^{m \, n}\) applies here.- The term \(4^2\) simplifies to \(16\) because \(4^2 = 16\).

Step 4: Combine the Simplified Terms

Combining the simplified terms from the previous step gives us:\[x^{4a} - 16\]This is the final expression after multiplying the given binomials.

Key Concepts

Understanding Binomials

When dealing with algebraic expressions, one common term you might come across is **binomials**. Binomials are simply algebraic expressions that contain two distinct terms. In our example,

• \((x^{2a} + 6)\) and
• \((x^{2a} - 4)\)

are the two binomials in the problem. Each of these expressions is formed by connecting two terms using a plus or minus sign.

The idea of **difference of squares** plays a crucial role in dealing with binomials. It is specifically applicable when your binomials look like \((a + b)(a - b)\), where the **first** and **second** terms are the same, except for their signs. This makes simplifying and solving products of such binomials straightforward using the formula:\[a^2 - b^2\].

Once you identify the terms (in our case, \(a = x^{2a}\) and \(b = 4\)), the simplification becomes more manageable. Keep this in mind as it will always help simplify binomial multiplication.

Exploring Exponent Rules

Understanding exponent rules is fundamental when working with algebraic expressions containing powers. Exponents tell you how many times a number, known as the base, is multiplied by itself. There are several rules that help simplify expressions with exponents:

• Product of Powers Rule: \(x^m \cdot x^n = x^{m+n}\), where the exponents are added if the base is the same.
• Power of a Power Rule: \((x^m)^n = x^{m \cdot n}\), where you multiply the exponents.
• Power of a Product Rule: \((xy)^n = x^n \cdot y^n\), where the exponent applies to each factor.

In the given problem, the expression \((x^{2a})^2\) uses the **Power of a Power Rule**. This simplifies the term to \(x^{4a}\) because multiplying \(2a \times 2\) gives \(4a\).

By mastering these rules, simplifying complex algebraic expressions becomes much more manageable.

Steps towards Simplification

**Simplification** in algebra means reducing an expression to its simplest form while keeping the value unchanged. This process can make equations easier to interpret and solve. In our exercise, simplifying the product of two binomials involves several key steps:

1. **Identify and Apply the Formula**: Recognize the given expression as a difference of squares, \((a + b)(a - b) = a^2 - b^2\).
2. **Substitute Known Values**: Substitute \(a = x^{2a}\) and \(b = 4\) into the formula to get \((x^{2a})^2 - 4^2\).
3. **Simplify Each Term**:
- The term \((x^{2a})^2\) becomes \(x^{4a}\) by applying exponent rules.
- The term \(4^2\) simply gives \(16\).
4. **Combine the Terms**: Gather the simplified terms together, resulting in \(x^{4a} - 16\).

By following these steps, you not only simplify the expression but also develop a deeper understanding of algebraic manipulation. This skill proves valuable in more complex mathematical tasks.