Problem 87 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The product is \( 6x^{3n+2} \).

Step 1: Separate the Coefficients and Variables

First, let's separate the coefficients and the variables. We have the expression \( (x^n)(2x^{2n})(3x^2) \), which can be rewritten as \( (1)(2)(3) \times (x^n)(x^{2n})(x^2) \).

Step 2: Multiply the Coefficients

Now, multiply the coefficients 1, 2, and 3. The multiplication is straightforward: \( 1 \times 2 \times 3 = 6 \).

Step 3: Add the Exponents of x

Next, we need to multiply the variables by adding their exponents. For the expression \( (x^n)(x^{2n})(x^2) \), we add the exponents: \( n + 2n + 2 \). This gives us \( 3n + 2 \).

Step 4: Combine Coefficients and Variables

Finally, combine the result from multiplying the coefficients with the result of adding the exponents. The complete expression is \( 6x^{3n+2} \).

Key Concepts

Exponents

Exponents are a way to indicate how many times a number, called the base, is multiplied by itself. When you see something like \( x^n \), it means 'x' is multiplied 'n' times. Exponents make expressions more concise and are used extensively in mathematics, especially in algebra.
Consider variables as placeholders for numbers. So, when a variable is involved, like \( x^n \), the 'n' acts as an exponent, telling you how many times 'x' will appear in a product.
To combine exponents with the same base, you add them. For example, with \( x^a \times x^b \), you will get \( x^{a+b} \). This is a fundamental rule when working with exponents in expressions.

Coefficients

Coefficients are the numbers that multiply variables in algebraic terms. For instance, in \( 3x^2 \), the number 3 is the coefficient. It describes how many sets of \( x^2 \) you have.
When you multiply expressions with the same or different coefficients, you simply multiply these numbers together. For example, if you have the expression \( 2(x^n) \) and \( 3(x^m) \), you multiply the coefficients to get \( 6(x^{n+m}) \). Simplifying by multiplying coefficients is often the first step in solving algebraic expressions that involve multiple terms.

Variable Multiplication

Variable multiplication involves combining variables, often by using exponents. In any given expression like \( (x^a)(x^b) \), the variables multiply based on their bases and exponents.
Importantly, make sure that the bases are the same before adding the exponents. For example, in the expression \( (y^1)(y^2) \), the base is 'y', so you add 1 and 2 to get \( y^3 \).
If you have different bases, such as \( (x^1)(y^2) \), you keep the variables separate since their bases do not match. This illustrates the rule that only like bases can have their exponents added.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations like addition and multiplication. They can range from simple, like \( x + 2 \), to complex, such as \( 3x^2 - 4xy + 7 \).
You often work with algebraic expressions to simplify or solve for unknown variables. To improve mastery, start by identifying different parts of the expression, such as individual terms and like terms.
In the exercise given, the algebraic expression \( (x^n)(2x^{2n})(3x^2) \), consists of three terms. Each term has a variable in the form of an exponent and a coefficient, making it a perfect example to practice combining like terms and simplifying complex expressions.