Problem 83 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

-10x^{2n}

Step 1: Identify Like Bases

Identify that the base in both terms of the expression \((2x^n)(-5x^n)\) is \(x\).

Step 2: Apply the Product of Powers Property

Use the product of powers property, which states \(a^m \cdot a^n = a^{m+n}\). Here, apply it to \(x^n \cdot x^n\), resulting in \(x^{n+n} = x^{2n}\).

Step 3: Multiply the Coefficients

Multiply the numerical coefficients: \(2\) and \(-5\). Thus, \(2 \cdot -5 = -10\).

Step 4: Combine the Results

Combine the results from steps 2 and 3 to write the complete expression: \(-10x^{2n}\). This is the product of the two expressions.

Key Concepts

Product of Powers Property

When working with algebraic expressions, especially those involving exponents, a helpful rule is the Product of Powers Property. This property states that when you multiply two powers that have the same base, you can simply add the exponents together. This can be represented by the formula: \[ a^m \cdot a^n = a^{m+n} \] where \( a \) is the common base and \( m \) and \( n \) are the exponents.
Here's a simple breakdown:

• Identify the base that is common between the two terms.
• Add their exponents together.

This property simplifies expressions significantly and is essential for solving many algebraic problems. For example, using this property, \(x^n \cdot x^n\) becomes \(x^{n+n} = x^{2n}\). See how easy that is?

Exponents

Exponents are a mathematical way to express repeated multiplication. For instance, \(x^n\) means that the base \(x\) is multiplied by itself \(n\) times. Exponents have many properties that make them versatile in algebra. Understanding these properties can help solve complex expressions more easily.
Some quick facts about exponents include:

• When multiplying like bases, you add the exponents (as in the Product of Powers).
• When dividing, you subtract the exponents.
• An exponent of zero means the expression equals one, assuming the base is not zero.

Knowing how exponents work aids in tackling various problems and simplifying expressions efficiently. Remember, handling exponents correctly can greatly affect the outcome of computations.

Multiplying Coefficients

When dealing with algebraic expressions that include coefficients and exponents, it's important to correctly handle the coefficients. Coefficients are the numbers that appear in front of the variable with an exponent. Multiplying coefficients follows basic arithmetic rules.
Here's how you can effectively multiply coefficients:

• Take the coefficients from each term and multiply them.
• Include the result as the new coefficient in your product.

Consider the expression \((2x^n)(-5x^n)\). The coefficients are \(2\) and \(-5\). Multiplying them together gives \(2 \cdot -5 = -10\). This product then merges with the result from multiplying the exponents, leading to the final expression. Properly handling coefficients ensures accurate mathematical solutions.