Problem 76 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The product is \(3x^{5n-1}\).

Step 1: Apply the Product of Powers Property

We are multiplying two expressions with a common base, which is the variable 'x'. The Product of Powers Property states that when multiplying powers with the same base, you add their exponents. This means we will add the exponents of the variable 'x' from each factor. So for the expression \((x^{2n})(x^{3n-1})\), add the exponents \(2n\) and \(3n-1\).

Step 2: Simplify the Exponents

Add the exponents from the two powers of 'x': \[2n + (3n - 1) = 2n + 3n - 1 = 5n - 1.\] This means the simplified exponent for 'x' is \(5n - 1\).

Step 3: Multiply the Numerical Coefficient

The first expression contains a numerical coefficient of 3. Since the second expression does not have a numerical coefficient other than the implied 1, you simply multiply 3 by 1. Thus, the product of the coefficients is 3.

Step 4: Write the Final Product

Combine the results from the previous steps. The numerical coefficient is 3, and the power of 'x' is \(x^{5n-1}\). Therefore, the final product is: \[3x^{5n-1}.\]

Key Concepts

Product of Powers Property

When you're multiplying powers that have the same base, it's helpful to use the Product of Powers Property. This rule tells us that when two exponential expressions with the same base are multiplied, the exponents can simply be added together. For instance, in the expression \((x^{a})(x^{b})\), the base is 'x' for both terms, and you would add the exponents: \(a + b\).
Using this property helps simplify expressions and keep calculations straightforward. It’s like stacking similar items—you combine them and count as you go along, which is a handy way to simplify work with algebraic expressions before proceeding to further calculations.
The Product of Powers Property is foundational in algebra and knowing how to use it efficiently is crucial for successfully manipulating expressions.

Exponent Rules

Exponent rules are key to making algebraic expressions more manageable. They consist of several properties, including the Product of Powers Property.

• Adding Exponents: This involves adding the powers when multiplying like bases.
• Zero Exponent Rule: Any base raised to the power of zero equals one, e.g., \(x^0 = 1\).
• Negative Exponent Rule: A negative exponent indicates a reciprocal, so \(x^{-a} = \frac{1}{x^a}\).

These rules also offer a shortcut to simplifying expressions, making calculations less cumbersome. Mastering them can transform algebra into a set of easy-to-follow steps rather than a complex maze. Always keep in mind that these rules aim to simplify and clarify your work with exponents.

Multiplying Expressions

When multiplying expressions, especially algebraic ones, you often encounter constants (numerical coefficients) along with variables raised to powers.
Consider the expression \((3x^{2n})(x^{3n-1})\). First, focus on the numbers: a 3 and the implied 1 next to the \(x^{3n-1}\). Multiplying these gives you just 3, since any number times 1 remains unchanged.
Next, deal with the variables: apply the Product of Powers Property by adding the exponents. Simplifying exponents streamlines the multiplication process, making it quicker and error-free. Finally, combine the numerical coefficient with the variable and its new exponent to write the final answer.
Multiplying expressions is often the first step in more complex calculations, so understanding it is essential for tackling bigger algebraic projects.