Problem 17 - Algebra for College Students

Question

Find each product. $$\left(-\frac{3}{4} a b\right)\left(\frac{1}{5} a^{2} b^{3}\right)$$

Step-by-Step Solution

Answer

The product is \(-\frac{3}{20}a^3b^4\).

Step 1: Multiply the Coefficients

Multiply the coefficients of the two terms. The coefficients are \(-\frac{3}{4}\) and \(\frac{1}{5}\). The product is \(-\frac{3}{4} \times \frac{1}{5} = -\frac{3}{20}\).

Step 2: Multiply the Variables with the Same Bases

For the variable \(a\), multiply \(a\) from the first term with \(a^2\) from the second term. This gives \(a^{1+2} = a^3\). For the variable \(b\), multiply \(b\) from the first term with \(b^3\) from the second term. This gives \(b^{1+3} = b^4\).

Step 3: Combine the Results

Combine the results from the previous steps to get the final product. The coefficient is \(-\frac{3}{20}\) and the products of the variables are \(a^3\) and \(b^4\). Thus, the final product is \(-\frac{3}{20}a^3b^4\).

Key Concepts

Understanding Coefficients in Algebraic Multiplication

When dealing with algebraic expressions, coefficients play a crucial role. A coefficient is the numerical part of a term that multiplies the variable(s). In the given problem, the coefficients are \(-\frac{3}{4}\) and \(\frac{1}{5}\). To find the product of an expression, multiply these coefficients together. This means you ignore the variables for a moment and focus solely on the numbers. In this problem, multiplying \(-\frac{3}{4}\) by \(\frac{1}{5}\) gives

• Multiply the numerators: \(-3\times1 = -3\)
• Multiply the denominators: \(4\times5 = 20\)

Thus, the product of the coefficients is \(-\frac{3}{20}\). Be careful with negative signs: they can change the sign of the product. Remember, a negative times a positive is always negative.

Exploring Variable Exponents

Variables in algebraic terms often have exponents, which indicate how many times the variable is multiplied by itself. When multiplying variables with exponents, you apply the rule of adding exponents if the bases are the same. In the expression, we have terms with \(a\) and \(b\). For variable \(a\):

• First term has \(a^1\)
• Second term has \(a^2\)
• Multiply them as \(a^{1+2} = a^3\)

This works because you add the exponents when multiplying like bases. Similarly, for variable \(b\):

• First term has \(b^1\)
• Second term has \(b^3\)
• Multiply them as \(b^{1+3} = b^4\)

Understanding this rule makes handling algebraic expressions with exponents more straightforward.

Handling Negative Numbers in Products

Negative numbers can be tricky when performing multiplication. It's important to understand how they interact with other numbers. When a negative number is multiplied by a positive one, the product is negative, as seen in the multiplication of our coefficients in the earlier steps. Here’s what to keep in mind:

• Negative times Positive = Negative
• Negative times Negative = Positive
• Positive times Positive = Positive

In the context of our problem, the coefficient \(-\frac{3}{4}\) is negative and \(\frac{1}{5}\) is positive. Their product \(-\frac{3}{20}\) reflects this rule. Remember, the sign of the ultimate product in an equation depends on the combination of these basic multiplication rules. Thus, understanding negative values is crucial for the accuracy of algebraic solutions.