Problem 66 - Algebra for College Students

Question

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\left(-9 a^{-3} b^{-6}\right)\left(-12 a^{-1} b^{4}\right)\)

Step-by-Step Solution

Answer

The product is \(\frac{108}{a^4 b^2}\).

Step 1: Rewrite with Positive Exponents

First, rewrite each term with positive exponents. Remember that a negative exponent means that the base is on the wrong side of the fraction: \(-9a^{-3}b^{-6} = -9 \frac{1}{a^3} \frac{1}{b^6}\).Similarly, for the second term: \(-12a^{-1}b^4 = -12 \frac{1}{a} b^4 \).

Step 2: Multiply Coefficients and Bases of the Same Variable

Multiply the numerical coefficients and each corresponding base separately. The coefficients: \((-9) \times (-12) = 108\).For the bases:\(a^{-3} \times a^{-1} = a^{-3-1} = a^{-4} \) and \(b^{-6} \times b^{4} = b^{-6+4} = b^{-2} \).

Step 3: Rewrite with Positive Exponents

Be sure that all exponents are positive by rewriting the expression:\(108 a^{-4} b^{-2}\) can be rewritten as \(\frac{108}{a^4 b^2}\).

Step 4: Combine Steps

The product of the two terms simplified and with positive exponents only is:\(\frac{108}{a^4 b^2}\).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. When you see something like \( a^3 \), it simply means \( a \) multiplied by \( a \) again and again, three times in total: \( a \times a \times a \).

In the context of algebra, exponents are crucial as they help simplify expressions and make them easier to work with. They follow specific rules such as the product of powers rule and the power of a power rule, which allow us to manipulate expressions in various ways. Understanding these rules is essential for solving problems that involve exponents effectively.

A key property of exponents is that multiplying two powers with the same base means you add the exponents: \( a^m \times a^n = a^{m+n} \). This rule forms the basis for handling expressions with exponents in algebra.

Multiplication of Algebraic Terms

When multiplying algebraic terms, it's important to multiply both the numerical coefficients and the variables separately. You'll often encounter terms where you have to combine numbers and multiple variables with different exponents.

Here's a step-by-step approach:

• Multiply the numerical coefficients (the numbers in front of the variables) first. For example, with \( -9 \) and \( -12 \), their product is \( 108 \) because \( (-9) \times (-12) = 108 \).
• Next, focus on the variables that have the same base, applying the rule of adding the exponents. If you have terms like \( a^{-3} \times a^{-1} \), you add the exponents: \( a^{-3 + (-1)} = a^{-4} \).

By consistently applying these steps, you can handle even complex algebraic expressions smoothly.

Negative Exponents

Negative exponents can seem tricky at first, but they have a straightforward interpretation: they indicate a reciprocal.

For instance, a term like \( a^{-3} \) really means \( \frac{1}{a^3} \). This tells you that the base is on the bottom of a fraction multiplier. Understanding negative exponents is crucial for making expressions simpler and rewriting them using only positive exponents.

To convert a negative exponent into a positive one, you can use the rule: \( a^{-m} = \frac{1}{a^m} \). Moving terms between the numerator and the denominator flips the sign of the exponent, turning negative powers into positive ones. This technique is especially useful for simplifying cleanup calculations in algebra.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves rewriting expressions so that they only contain positive exponents. This can make them easier to understand and work with.

The process often involves:

• Converting all terms with negative exponents to positive by moving them across the fraction bar. For instance, \( a^{-4} \) becomes \( \frac{1}{a^4} \).
• Simplifying the expression by multiplying coefficients and adjusting exponents according to multiplication rules.
• Ensuring that you finally express your results clearly, using positive exponents to avoid unnecessary complexity.

By following this approach, you learn to break down complex algebraic expressions into simpler fractions, making them manageable without compromising clarity.