Problem 18 - Algebra for College Students

Question

Find each product. $$\left(-\frac{2}{7} a^{2}\right)\left(\frac{3}{5} a b^{3}\right)$$

Step-by-Step Solution

Answer

The product is \(-\frac{6}{35}a^3b^3\).

Step 1: Identify the Expression to Simplify

We need to find the product of two terms: \[\left(-\frac{2}{7} a^{2}\right)\left(\frac{3}{5} a b^{3}\right)\]The expression involves multiplying fractions and variables such as \(a^2\) and \(ab^3\).

Step 2: Multiply the Fractions

To multiply fractions, multiply the numerators and then the denominators:- Numerators: \((-2) \times 3 = -6\)- Denominators: \(7 \times 5 = 35\)So, the fraction part of the product is \[-\frac{6}{35}\]

Step 3: Multiply the Like Variables

For the variables, multiply them by adding their exponents if they have the same base:- \(a^2\) and \(a\) multiply to give \(a^{2+1} = a^3\).- When multiplying \(a^2\) with \(b^3\), notice that there is no \(b\) in the first term, so it remains\(b^3\).The variable part of the product is \(a^3b^3\).

Step 4: Combine the Results

Combine the results from Step 2 and Step 3:The final expression combining both the fraction and the variables is:\[-\frac{6}{35}a^3b^3\]

Key Concepts

Multiplying Fractions

To multiply fractions, you need to focus on two main aspects: the numerators and the denominators. Each fraction can be broken down into these two parts. When multiplying fractions, follow these steps:

• Multiply the numerators. This means you take the top numbers of the fractions and multiply them together.
• Multiply the denominators. This means you take the bottom numbers of the fractions and do the same.

For the given problem, we multiplied \(-2\) and \(3\) to get \(-6\) for the numerators. For the denominators, we multiplied \(7\) and \(5\) which gave \(35\). Therefore, the resulting fraction from this multiplication step is \(-\frac{6}{35}\). This method keeps multiplication straightforward, ensuring you handle both parts of the fraction separately until you're ready to combine them into one new fraction.

Exponents in Algebra

Working with exponents is all about understanding their fundamental properties, especially when multiplying variables. An exponent shows how many times a number, or a variable, is multiplied by itself. Let's break down the steps:

• If you are multiplying variables with the same base, you need to add the exponents together.
• Remember that different variables do not combine exponents unless they share the same base.

In the example given, we see two terms: \(a^2\) and \(a\). Here, they share the base \(a\), allowing you to add the exponents: \(2 + 1\). This results in \(a^3\). However, the base of \(b^3\) does not combine with \(a\) or its exponents because their bases are different. Thus, \(b^3\) remains unchanged. Keeping track of bases and knowing this rule of adding exponents is vital for handling polynomial expressions smoothly.

Polynomial Multiplication

Polynomial multiplication brings together both fractional multiplication and the rules of exponents. When multiplying polynomials, consider each part separately. Let's overview the process:

• Multiply the coefficients. These are the numerical parts of the terms, usually found in front of the variables.
• Multiply the variables using the rules of exponents, as outlined above.

Applying this to our exercise, the coefficients \(-\frac{2}{7}\) and \(\frac{3}{5}\) multiplied to form \(-\frac{6}{35}\). For the variables, \(a^2\) multiplied with \(a\) yields \(a^3\), and \(b^3\) remains as it is since there is no \(b\) in the first term. The finished expression \(-\frac{6}{35}a^3b^3\) merges these separate calculations into a single polynomial. Understanding how to multiply both the constants and the variables is key when tackling any polynomial multiplication.