Problem 77 - Algebra for College Students

Question

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(a^{2 n-1}\right)\left(a^{2 n+4}\right)$$

Step-by-Step Solution

Answer

The product is \(a^{4n+3}\).

Step 1: Identify the Base

Look at the given expression: \(\left(a^{2n-1}\right)\left(a^{2n+4}\right)\). Notice that the base of the expression is \(a\). Both terms have the same base, which allows us to apply the exponent rule for multiplication.

Step 2: Apply the Exponent Multiplication Rule

When multiplying terms with the same base, we add the exponents together. The general rule is \(a^m \cdot a^n = a^{m+n}\). Here, \(m = 2n - 1\) and \(n = 2n + 4\).

Step 3: Add the Exponents

We add the exponents \(2n - 1\) and \(2n + 4\) to get the new exponent for the product of the bases. Perform the addition: \[2n - 1 + 2n + 4 = 4n + 3\]

Step 4: Combine the Base and New Exponent

Combine the base \(a\) with the new exponent \(4n + 3\). The result is the product: \[a^{4n+3}\]

Key Concepts

Exponent Rules

Exponent rules are essential in algebra as they help in simplifying expressions involving powers. One of the fundamental rules is the product rule, which applies when multiplying terms with the same base. The rule states: \(a^m \cdot a^n = a^{m+n}\). This formula means that you keep the same base and simply add the exponents.

Applying this rule allows us to simplify expressions efficiently. For instance, if you have \(b^2 \cdot b^3\), you add the exponents (2 and 3) to get \(b^5\). This is because the base \(b\) remains unchanged while multiplying, and you are essentially multiplying \(b\) by itself a total of five times.

• Always check that the bases are the same before applying this rule.
• Ensure that the exponents, which may include variables, are simply added together.

Understanding exponent rules streamlines working with polynomials by ensuring each step is logical and correct.

Variable Expressions

Variable expressions consist of numbers, variables, and operators. They are used widely in algebra to represent unknown values or quantities. In the expression \(2x + 3\), for example, 'x' is the variable meaning it can take various values.

When dealing with variables, it is important to be consistent with their use. Variables are treated like numbers when performing mathematical operations like addition, subtraction, multiplication, and division.

In particular, look out for

• Combining like terms: Terms that contain the same variable raised to the same power. For example, \(3x^2 + 2x^2\) can be simplified to \(5x^2\).
• Substitution: Replacing variables with actual numbers in some cases to evaluate expressions.

Knowing how to handle variables and their expressions can help solve complex equations and understand the behavior of different functions.

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials, which can be simpler when understood step-by-step. It can include multiplying monomials, binomials, or even larger polynomials.

To multiply polynomials, use the distributive property, which states \((a+b)(c+d)=ac+ad+bc+bd\). This means you multiply each term in the first polynomial with every term in the second polynomial.

Here's a methodical approach:

• Write each polynomial in standard form with terms in descending order of exponents.
• Multiply the coefficients and add the exponents of like bases.
• Combine like terms in the resulting expression.

For example, multiplying \((x+3)(x+2)\) leads to \(x^2 + 5x + 6\) after distributing each term and combining like terms. Practicing polynomial multiplication helps in understanding more advanced algebraic concepts and prepares you for calculus.