Problem 68 - Algebra for College Students

Question

Find the indicated products and quotients. Express final results using positive integral exponents only. \(\frac{-72 a^{2} b^{-4}}{6 a^{3} b^{-7}}\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{12 b^{3}}{a}\).

Step 1: Simplify the coefficients

Begin by dividing the numerical coefficients: \[\frac{-72}{6} = -12\]So the simplified coefficient is \(-12\).

Step 2: Simplify the powers of \(a\)

Apply the quotient of powers rule: for the bases that are same, subtract the exponents. For base \(a\):\[a^{2} \div a^{3} = a^{2-3} = a^{-1}\]This simplifies to \(a^{-1}\).

Step 3: Simplify the powers of \(b\)

Apply the quotient of powers rule for base \(b\):\[b^{-4} \div b^{-7} = b^{-4 - (-7)} = b^{3}\]This simplifies to \(b^3\).

Step 4: Combine the results

Combine all the simplified parts together. The expression simplifies to:\[-12 \cdot a^{-1} \cdot b^{3}\]Since we need to express with positive exponents, we rewrite \(a^{-1}\) as \(\frac{1}{a}\):\[-12 \cdot \frac{b^{3}}{a}\]Thus, the final answer is:\[-\frac{12 b^{3}}{a}\]

Key Concepts

Exponents

Exponents play a crucial role in algebra, especially when it comes to expressions involving repeated multiplication. An exponent indicates how many times you use a number, known as the base, in a multiplication. For example, in the expression \(a^2\), the base \(a\) is multiplied by itself, resulting in \(a \times a\). This tells us the power or degree of the base.

Exponents follow specific rules and properties that help simplify mathematical expressions and equations. These rules include the product of powers, power of a power, and more. Different bases and their exponents must be treated properly during calculations to ensure the correct solution, as seen in exercises like this one.

Quotient of Powers Rule

The Quotient of Powers Rule is one of the essential rules when dealing with exponents. This rule states that when you divide two powers with the same base, you simply subtract the exponents. This means that for any nonzero base \(x\), the expression \(\frac{x^m}{x^n} = x^{m-n}\).

Let's see how this rule applies. In our exercise, we have \(a^2 \div a^3\). By applying the rule, you subtract the exponents: \(2 - 3 = -1\), resulting in \(a^{-1}\). Note that when the exponent becomes negative, it indicates a reciprocal; \(a^{-1}\) can be rewritten as \(\frac{1}{a}\).

This rule simplifies expressions significantly, especially when working with several variable terms in the same division.

Simplifying Expressions

Simplifying expressions is a fundamental aspect of algebra. It involves using various algebraic rules to reduce expressions to their simplest form, making them easier to understand and work with. This process often includes the application of rules for exponents and aims to express all results using positive exponents.

In the given exercise, simplification is done step by step. First, the numerical coefficients are simplified by dividing them, simplifying \(-72 \div 6\) to \(-12\). Then, the quotient of powers rule is applied separately to the variables \(a\) and \(b\), resulting in the terms \(a^{-1}\) and \(b^3\), respectively. Lastly, the expression is combined and rewritten with positive exponents: \(-\frac{12b^3}{a}\).

By mastering these simplification techniques, you'll not only find it easier to solve complex algebraic problems, but it will also enhance your overall mathematical understanding.