Problem 11 - Algebra for College Students

Question

Find each product. $$\left(3 a^{2} b\right)\left(9 a^{2} b^{4}\right)$$

Step-by-Step Solution

Answer

The product is \( 27a^4b^5 \).

Step 1: Understand the Problem

The task is to find the product of two algebraic expressions: \( (3a^2b) \) and \( (9a^2b^4) \). We will multiply these expressions together by combining their coefficients and variables.

Step 2: Multiply the Coefficients

First, identify the coefficients in each expression. The coefficients are 3 and 9. Multiply these coefficients together: \( 3 \times 9 = 27 \).

Step 3: Multiply the Variables of Same Type

Now, multiply the variable parts of the expressions. Start with \( a^2 \) from both terms. Multiply them using the law of exponents: \( a^2 \cdot a^2 = a^{2+2} = a^4 \).

Step 4: Multiply the Second Set of Variables

Next, take the variable \( b \) terms from both expressions. Multiply \( b \) with \( b^4 \): \( b \cdot b^4 = b^{1+4} = b^5 \).

Step 5: Combine Results

Finally, combine the results from Step 2, Step 3, and Step 4: \( 27 \cdot a^4 \cdot b^5 = 27a^4b^5 \). This is the product of the two expressions.

Key Concepts

Law of Exponents

When working with algebraic expressions, understanding the law of exponents is crucial. It helps in efficiently multiplying variables. This law states that when you multiply two expressions that have the same base, you can add the exponents.
For example, if you have two terms such as \( a^m \) and \( a^n \), then \( a^m \cdot a^n = a^{m+n} \). This property simplifies the process of dealing with powers in expressions.
In our exercise, we used this law to multiply \( a^2 \cdot a^2 \), resulting in \( a^{2+2} = a^4 \), showing how simple the rule can apply to streamline calculations.

• This rule only applies when multiplying terms with the same base.
• Always ensure that the bases are identical before applying this rule.

Multiplying Coefficients

Coefficients in algebraic expressions are the numbers that appear in front of the variables. They can be thought of as the amounts that scale the variables. When multiplying two expressions, it's important to first handle the coefficients correctly.
In the example \( (3a^2b) \cdot (9a^2b^4) \), we identified 3 and 9 as coefficients. These are simply multiplied together just like regular numbers: \( 3 \times 9 = 27 \).
This step simplifies the expressions and prepares us for the next steps, where the variable parts are multiplied according to their respective laws.

• Always identify and multiply coefficients first when dealing with expression multiplications.
• Coefficients are treated as normal numbers; focus on these before handling variable terms.

Variables in Algebra

Variables represent unknown or changeable values in algebra. They are often denoted by letters such as \( a \) or \( b \). Proper handling of variables is essential for algebraic operations.
In an expression like \( 3a^2b \), \( a^2 \) and \( b \) are variable components, each raised to a certain power. These powers dictate how they interact with variables of the same type when multiplied or divided.
When multiplying expressions, identify variables with the same base and apply the laws of exponents. For example, multiplying \( b \cdot b^4 \), we add the exponents to obtain \( b^{1+4} = b^5 \).

• Variables allow flexibility and expression in equations – they can symbolize any number.
• Understand the role of variables and their exponents in an expression to master algebraic manipulation.