Problem 6 - Algebra for College Students

Question

Find each product. $$\left(-8 a^{2} b^{2}\right)\left(-3 a b^{3}\right)$$

Step-by-Step Solution

Answer

The product is \(24a^3b^5\).

Step 1: Identify Like Terms

The given expression is \( (-8a^2b^2)(-3ab^3) \). First, identify like terms to combine. Notice that both parts contain powers of \(a\) and \(b\).

Step 2: Multiply Coefficients

Multiply the coefficients \(-8\) and \(-3\). This gives: \(-8 \times -3 = 24\).

Step 3: Apply the Product Rule to \(a\) terms

For the \(a\) terms, use the exponent multiplication rule \(a^m \cdot a^n = a^{m+n}\). Here, \( a^2 \cdot a^1 = a^{2+1} = a^3 \).

Step 4: Apply the Product Rule to \(b\) terms

For the \(b\) terms, apply the exponent multiplication rule \(b^m \cdot b^n = b^{m+n}\). Here, \( b^2 \cdot b^3 = b^{2+3} = b^5 \).

Step 5: Combine All Parts

Combine the results from all previous steps: The product is \(24a^3b^5\).

Key Concepts

Product Rule for Exponents

When you encounter powers of the same base being multiplied in algebra, the Product Rule for Exponents is your go-to tool. This rule states that when you multiply similar bases, you simply add their exponents. Here, you're working with variables like \(a\) and \(b\). To apply the rule:

• Identify the bases that are the same, such as \(a\) in \(a^2\) and \(a\), and \(b\) in \(b^2\) and \(b^3\).
• Add the exponents of each base: For \(a\), transform \(a^2 \times a^1\) into \(a^{2+1} = a^3\). For \(b\), change \(b^2 \times b^3\) into \(b^{2+3} = b^5\).

By reorganizing like this, you ensure your expression follows proper mathematical rules, resulting in accurate calculations. Utilizing this rule effectively simplifies problems involving exponents, bringing clarity and precision to your solutions.

Coefficient Multiplication

In algebra, coefficients are the numerical factors in terms, and multiplying them is straightforward yet essential. Coefficients are the numbers directly in front of a variable or a set of variables. In this exercise, you're dealing with coefficients -8 and -3. Follow these steps:

• Multiply the coefficients together: \(-8\times-3\).
• Calculate the product: Multiply the absolute values, \(8\times3=24\), then consider the signs. Both numbers are negative, so the result is positive.

Thus, the multiplication of coefficients yields \(24\). Remember that handling the coefficients separately from the variables often simplifies your work, and correctly addressing sign changes is crucial.

Combining Like Terms

Combining like terms means bringing together terms with the same variables raised to the same powers. It simplifies expressions, making them easier to work with. In our exercise, after applying the product rule and multiplying coefficients, you're left with:

• \(24a^{3}\)
• \(b^{5}\)

Since there are no other terms with these exact variables and exponents, you simply write them out together, resulting in \(24a^{3}b^{5}\). Always ensure all terms have been reduced according to their similarities. If variables and exponents match, combine them. If not, list them in the product or sum. This approach streamlines expressions and ensures clarity throughout your calculations.