Problem 8 - Elementary Algebra

Question

Perform each multiplication in one step. $$ 4 a^{3} b^{2} \cdot 9 a^{2} b $$

Step-by-Step Solution

Answer

Question: Multiply the expressions \(4a^3b^2\) and \(9a^2b\). Answer: After multiplying the given expressions, the result is \(36a^5b^3\).

Step 1: Identify the multiplication of constants

Now, we will multiply the constants (numbers) in the given expression. Here, 4 and 9 are the constants. $$ 4 \cdot 9 = 36 $$

Step 2: Identify and multiply coefficients with the same base (a)

We will now multiply the coefficients with the same base, in this case 'a'. We have \(a^3\) and \(a^2\). Using the property of exponents (\((a^m)^n = a^{m \times n}\)), we get: $$ a^3 \cdot a^2 = a^{3+2} = a^5 $$

Step 3: Identify and multiply coefficients with the same base (b)

Next, we multiply coefficients with the same base, in this case, 'b'. We have \(b^2\) and \(b^1\) (as just 'b' can be considered as \(b^1\)). Applying the property of exponents (\((a^m)^n = a^{m \times n}\)) again, we have: $$ b^2 \cdot b^1 = b^{2+1} = b^3 $$

Step 4: Combine the results from Steps 1, 2, and 3

Combine the constants and terms from steps 1, 2, and 3 to get the final answer. $$ 36 \cdot a^5 \cdot b^3 $$ So, the result of the multiplication is: $$ 36 a^5 b^3 $$

Key Concepts

Understanding Exponents

Exponents are a fundamental aspect of mathematics and are crucial when dealing with polynomials and algebraic expressions. An exponent refers to the number of times a base number is multiplied by itself. For instance, in the expression \(a^3\), the 'a' is the base and '3' is the exponent. This indicates that 'a' is multiplied by itself three times: \(a \cdot a \cdot a\).

When multiplying numbers with the same base, we add their exponents. For the expression \(a^3 \cdot a^2\), both terms have the base 'a'. Applying the rule for exponents, their powers are summed as follows:

• \(a^{3+2} = a^5\)

This quick rule streamlines calculations, especially within larger expressions. Grasping and applying these properties of exponents makes solving algebraic problems simpler and more intuitive.

Exploring Expressions

Expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication, etc.) that help us convey mathematical relationships. An algebraic expression does not have an equal sign, distinguishing it from an equation. An example of an expression is \(4a^3b^2\), which includes both numerical and variable components.

In polynomial multiplication, like the expression \(4a^3b^2 \cdot 9a^2b\), each part of the expression contributes to the overall calculation:

• Numerical values like '4' and '9' are constants.
• Variables 'a' and 'b' are combined according to algebraic rules.

By understanding each component's role, students can break down and solve complex polynomial multiplication with ease.

Defining Monomials

A monomial is a specific type of expression noted for its simplicity; it consists of a single term. Each monomial is typically a product of constants and variables raised to whole-number exponents. Examples of monomials include \(5x\), \(7y^2\), and \(3ab^3\).

In mathematical expressions, monomials can be multiplied together via straightforward rules focusing on exponent addition and constant multiplication. For instance, multiplying the monomials \(4a^3b^2\) and \(9a^2b\) involves:

• Multiplying constants: \(4 \cdot 9 = 36\)
• Adding exponents for 'a': \(a^3 \cdot a^2 = a^{3+2} = a^5\)
• Adding exponents for 'b': \(b^2 \cdot b^1 = b^{2+1} = b^3\)

By understanding these basic operations, students gain the ability to solve polynomials efficiently and accurately.