Problem 75 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The product is \(6x^{3n}\).

Step 1: Identify the terms in the product

Look at the given expression: \((2x^n)(3x^{2n})\). We have two coefficients, 2 and 3, and variables with exponents,\(x^n\) and \(x^{2n}\).

Step 2: Multiply the coefficients

Multiply the numerical coefficients: \(2\) and \(3\). This gives us \(2 \times 3 = 6\).

Step 3: Apply the property of exponents

Use the property of exponents stating \(x^a \times x^b = x^{a+b}\). For the terms \(x^n\) and \(x^{2n}\), add the exponents: \(n + 2n = 3n\).

Step 4: Combine the results

Combine the result from multiplying the coefficients with the result from adding the exponents. We get\(6x^{3n}\).

Step 5: Present the final product

Thus, the product of \((2x^n)(3x^{2n})\) is \(6x^{3n}\).

Key Concepts

Multiplication of Algebraic Expressions

When multiplying algebraic expressions, it's crucial to carefully handle both the coefficients and the variables. In the expression \((2x^n)(3x^{2n})\), the coefficients are the numerical parts, \(2\) and \(3\), while the variables are the parts with exponents, \(x^n\) and \(x^{2n}\). The multiplication involves two steps:

• Multiplying the Coefficients: Begin by multiplying the numbers: \(2\times 3\), which results in \(6\).
• Dealing with the Variables: When the variables have exponents, as in \(x^n\times x^{2n}\), you apply the rules of exponents to combine them.

By following these steps, the expression is simplified to a single term that combines both numerical and variable components, resulting in \(6x^{3n}\). Understanding this process aids in dealing with more complex algebraic expressions effectively.

Properties of Exponents

Exponents indicate how many times a number, known as the base, is multiplied by itself. In our example, this is the \(x\) in both \(x^n\) and \(x^{2n}\). When multiplying terms with the same base, you can add the exponents according to the property \(x^a \times x^b = x^{a+b}\).

• This means for \(x^n\) and \(x^{2n}\), you simply add the exponents: \(n + 2n\), resulting in \(3n\).
• It's crucial to remember that this property only applies when the bases (\(x\)) are the same. If you encounter terms with different bases, the rule changes.

This exponent property is fundamental in algebra, providing a simplified method to handle what would otherwise be lengthy multiplication.

Integer Exponents

Integer exponents often appear in algebraic expressions, denoting repeated multiplication. These integers can be positive, indicating straightforward multiplication, or negative, indicating division or reciprocals. In our problem, both exponents, \(n\) and \(2n\), are positive integers. This simplifies the multiplication process using the rules of exponents.

• Larger exponents signify that the base number is multiplied by itself more times. For \(x^{3n}\), this means \(x\) is multiplied by itself \(3n\) times.
• When combining expressions like this, it's essential to handle exponents correctly to ensure accurate results.

Being comfortable with integer exponents helps tackle various algebraic operations confidently, as they frequently appear in both simple and complex equations.