Problem 44 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ (3 a b)\left(2 a^{2} b\right) $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((3ab)(2a^2b)\). Answer: The simplified expression is \(6a^3b^2\).

Step 1: Rewrite the expression

Rewrite the given expression \((3ab)(2a^2b)\), so it is easier to work with. $$ (3ab)(2a^2b) = 3 \cdot 2 \cdot a \cdot a^2 \cdot b \cdot b $$

Step 2: Apply the product rule for exponents

Now, we need to apply the product rule for exponents, which tells us that when multiplying two exponential expressions with the same base, we add the exponents. In this case, we have two sets of exponents with the same base (a and b): $$ a^1 \cdot a^2 = a^{1+2} = a^3 $$ and $$ b^1 \cdot b^1 = b^{1+1} = b^2 $$

Step 3: Simplify the expression

Now, let’s replace the new exponential expressions in the original expression and multiply the coefficients (numerical values): $$ 3 \cdot 2 \cdot a^3 \cdot b^2 = 6a^3b^2 $$ So, the simplified expression is: \(6a^3b^2\).

Key Concepts

Product Rule

When you multiply two exponential expressions that have the same base, a useful tool to utilize is the product rule. According to this rule, you simply need to add the exponents together. For example, if you have \(a^m \cdot a^n\), by applying the product rule, you get \(a^{m+n}\). This step occurs because you are dealing with repeated multiplication of the same factor when combining like bases.

In the original problem we are discussing, notice how there's \(a\) and \(b\) in both parts of the expression \((3ab)(2a^2b)\). We apply the product rule to these bases:

• For \(a\), it's \(a^1 \cdot a^2 = a^{1+2} = a^3\).
• For \(b\), it's \(b^1 \cdot b^1 = b^{1+1} = b^2\).

This makes it simpler and results in a cohesive expression.

Quotient Rule

The quotient rule is another key exponent rule. Although it was not directly used in simplifying the given problem, it's important to understand its function for complete learning. The quotient rule is the opposite of the product rule and is used when dividing two expressions with the same base. It states that you subtract the exponents instead of adding them.

For example, if given \(a^m \div a^n\), the quotient rule tells us to simplify to \(a^{m-n}\). Therefore, base division becomes managing the difference in powers. This concept is crucial when you encounter expressions that need further reduction.

• For instance, if you have \(x^5 \div x^2\), applying the quotient rule gives: \(x^{5-2} = x^3\).

Remembering both rules enables you to handle both multiplication and division problems efficiently.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves combining like terms and reducing expressions to their simplest form. This process often uses exponent rules such as the product and quotient rules to streamline computation.

In our given problem, you start with the expression, and by breaking it down (distributing or rearranging terms), you make it more manageable. Next, identify like terms, use the product rule on bases that are common, and finally multiply any numerical coefficients you've pulled out front.

• For example, \((3ab)(2a^2b)\) is broken down to \(3 \cdot 2 \cdot a \cdot a^2 \cdot b \cdot b\). Then, by applying rules for exponents, we reach \(6a^3b^2\).

Simplified expressions are easier to work with, especially when solving larger algebraic equations or systems.

Algebraic Expressions

An algebraic expression combines numbers, variables (like \(a\) or \(b\)), and arithmetic operations such as addition, subtraction, multiplication, and division. These expressions do not have an equal sign, which distinguishes them from equations.

In dealing with algebraic expressions, understanding how to manipulate and simplify them is key. They form the backbone of algebra and precalculus subjects, showcasing concepts like polynomial multiplication, distribution, and the aforementioned exponent rules.

• Take the expression \((3ab)(2a^2b)\) as given in this exercise: each component has a role, and through application of mathematical rules, we find an elegant, tidy expression: \(6a^3b^2\).

Learning to work with these expressions aids in problem-solving and predictive modeling in more advanced mathematics.