Problem 79 - Algebra for College Students

Question

Express each of the following as a single fraction involving positive exponents only. \(3 a^{-2}+4 b^{-1}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{3b + 4a^2}{a^2b}\).

Step 1: Identify the Denominators

The given expression is \(3a^{-2} + 4b^{-1}\). Here, terms involving negative exponents can be rewritten with positive exponents. We have \(a^{-2} = \frac{1}{a^2}\) and \(b^{-1} = \frac{1}{b}\). So the denominators of the fractions will be \(a^2\) and \(b\).

Step 2: Remove Negative Exponents

Convert the terms with negative exponents into fractions with positive exponents: \(3a^{-2} = \frac{3}{a^2}\) and \(4b^{-1} = \frac{4}{b}\). Now the expression is \(\frac{3}{a^2} + \frac{4}{b}\).

Step 3: Find the Least Common Denominator (LCD)

Identify the least common denominator for the fractions \(\frac{3}{a^2}\) and \(\frac{4}{b}\). The LCD is \(a^2b\).

Step 4: Express Each Term with the LCD

Rewrite each fraction with the least common denominator: \(\frac{3}{a^2} = \frac{3b}{a^2b}\) and \(\frac{4}{b} = \frac{4a^2}{a^2b}\).

Step 5: Add the Fractions

Add the fractions by combining them over the common denominator: \(\frac{3b}{a^2b} + \frac{4a^2}{a^2b} = \frac{3b + 4a^2}{a^2b}\).

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of two parts: the numerator and the denominator. The numerator is the top number and indicates how many parts are taken, while the denominator is the bottom number and indicates how many equal parts the whole is divided into. In algebra, fractions are essential for handling expressions that involve division.
Understanding how to manipulate fractions, especially when they involve variables, is crucial.

• Fractions can have variables in either the numerator, the denominator, or both.
• To add or subtract fractions, they must have the same denominator.
• The least common denominator (LCD) is the smallest number that is a multiple of both denominators.

Converting the expression into fractions allows for a unified way to handle different terms. This is why we rewrite terms with negative exponents as fractions, making it easier to combine them with a common denominator.

Exponents

Exponents are a mathematical notation that indicates how many times a number is multiplied by itself. An exponent is a small number written above and to the right of a base number. For example, in the expression \( a^2 \), \( a \) is the base and \( 2 \) is the exponent. This means \( a \) multiplied by itself: \( a \times a \). Exponents are used to simplify multiplication and express large numbers more succinctly.
In algebra, exponents help in dealing with polynomial expressions.

• Exponents follow specific rules, such as the product of powers rule: \( a^m \times a^n = a^{m+n} \).
• There’s also the power of a power rule: \((a^m)^n = a^{m \times n}\).

These rules form the foundation for manipulating expressions involving exponents, allowing us to transform and simplify complex algebraic expressions.

Negative Exponents

Negative exponents often confuse students, as they imply division rather than multiplication. A negative exponent indicates that the base is on the wrong level of a fraction and needs to be flipped to make it positive. For example, \( a^{-n} \) means \( \frac{1}{a^n} \). This transformation is crucial when working with different algebraic expressions, making them easier to manipulate and combine.
Here’s why understanding negative exponents is useful:

• They help convert tiny decimal numbers into whole numbers.
• Negative exponents in algebraic expressions are often rewritten as fractions to ease simplification and obtain a common denominator.
• Mastery of this leads to simplifying expressions and solving equations effectively.

By comprehending how to handle negative exponents, students can accurately simplify and solve complex algebraic expressions such as \( 3a^{-2} + 4b^{-1} \) by rewriting them as positive exponents and fractions and then proceeding with further algebraic operations.