Problem 65 - Algebra for College Students

Question

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(-7 a^{2} b^{-5}\right)\left(-a^{-2} b^{7}\right)\)

Step-by-Step Solution

Answer

The product is \( 7b^{2} \).

Step 1: Identify the Expression

We are given the expression \( \left(-7 a^{2} b^{-5}\right)\left(-a^{-2} b^{7}\right) \). We need to multiply these two expressions together.

Step 2: Multiply the Coefficients

First, we multiply the numerical coefficients: \(-7\) and \(-1\). Since \((-7)\times(-1) = 7\), the product of the coefficients is 7.

Step 3: Apply the Laws of Exponents to \(a\)

For the variable \(a\), we have \(a^{2} \times a^{-2}\). According to the laws of exponents, \(a^{m} \times a^{n} = a^{m+n}\). Therefore, \(a^{2} \times a^{-2} = a^{2-2} = a^{0}\). Since \(a^{0} = 1\), the \(a\) terms cancel out.

Step 4: Apply the Laws of Exponents to \(b\)

For the variable \(b\), we have \(b^{-5} \times b^{7}\). According to the laws of exponents, \(b^{m} \times b^{n} = b^{m+n}\). Therefore, \(b^{-5} \times b^{7} = b^{-5+7} = b^{2}\).

Step 5: Combine the Results

The product of the given expressions is obtained by combining the results from the previous steps. We have the coefficient 7, the \(a\) terms canceled out to 1, and \(b^{2}\). Therefore, the expression simplifies to \(7b^{2}\).

Key Concepts

Laws of Exponents

The laws of exponents are essential when dealing with algebraic expressions that involve powers. They provide rules for how to handle the multiplication and division of similar bases, making calculations much easier. Here are some of the key laws often used:

• Product of Powers: When multiplying powers that have the same base, you simply add the exponents. In mathematical terms, this is expressed as \(a^m \times a^n = a^{m+n}\). This rule is crucial when you encounter terms like \(a^2 \times a^{-2}\).
• Power of a Power: If you have a power raised to another power, multiply the exponents. For instance, \((a^m)^n = a^{m \times n}\).
• Zero Exponent Rule: Any base raised to the power of zero is 1, i.e., \(a^0 = 1\). This is particularly useful when the exponents combine to become zero during calculations.

Applying these laws to our expression, we found that multiplying \(a^2\) and \(a^{-2}\) resulted in \(a^0\), which equals 1, effectively canceling those terms out. For \(b\), applying the same product of powers rule resulted in \(b^{-5+7} = b^2\). Understanding these laws can greatly simplify algebraic operations.

Multiplication of Coefficients

In any algebraic expression, coefficients are the numerical parts that multiply the variables. When multiplying expressions, it's essential to first tackle these coefficients separately from the variables. Here's a breakdown:

• Identifying Coefficients: Look for numbers ahead of any variable or group of variables, like \(-7\) and \(-1\) in our example.
• Simple Multiplication: Multiply these numbers as you would with basic arithmetic: \((-7)\times(-1) = 7\). The sign rule for multiplication makes it clear that a negative times a negative gives a positive result.

So, by multiplying the coefficients separately, we achieved half the task with ease as numbers are more straightforward to handle. Following this rule ensures clarity and accuracy in larger, more complex expressions as well.

Simplification of Expressions

Simplifying an expression means reducing it to its most compact and understandable form without changing its value. It's a crucial skill in algebra to ensure your answers are both correct and presentable.

• Combine Like Terms: This involves adding or subtracting terms that have identical variables and exponents. For instance, \(b^{-5}\) and \(b^7\) were first combined using exponent laws to form \(b^2\).
• Use of Laws and Rules: Effective use of exponent laws often leads to significant simplification. In our example, applying the rules reduced the expression to \(7b^2\), eliminating any unnecessary parts like the cancelled \(a^0 = 1\).
• Final Check: Always double-check your simplification to ensure no errors were made during the process. Verify each applied rule for correctness, as simplicity should not compromise accuracy.

By following these strategies, you can distill even seemingly complex algebraic expressions to their essence, making them easier to understand and work with in further calculations or applications.