Problem 67 - Algebra 2 and Trigonometry

Question

In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ 12 a b \div 2 a b^{2} $$

Step-by-Step Solution

Answer

The quotient \(12ab \div 2ab^2\) simplifies to \(6b^{-1}\).

Step 1: Rewrite the Quotient

Given the expression \(12ab \div 2ab^2\), rewrite it in fraction form: \[ \frac{12ab}{2ab^2} \].

Step 2: Simplify the Coefficients

First, simplify the numerical coefficients. Divide the numerator 12 by the denominator 2: \[ \frac{12}{2} = 6 \].

Step 3: Simplify the Variable 'a'

The variable \(a\) in the numerator and the \(a\) in the denominator can be cancelled: \[ \frac{a}{a} = 1 \], leaving \(a\) to be fully cancelled out from both the numerator and denominator.

Step 4: Simplify the Variable 'b'

For the \(b\) terms, subtract the power from the denominator from the power in the numerator: \(b^{1 - 2} = b^{-1}\). This makes the term \(b^{-1}\), which is equivalent to \(\frac{1}{b}\).

Step 5: Rewrite as a Product

Combine the simplified terms to write the quotient as a product: \[ 6b^{-1} \]. This expression has no denominator.

Key Concepts

Simplifying Expressions

Simplifying expressions is an important mathematical process that helps to make complex algebraic formulas more manageable. When dealing with algebraic fractions, as seen in the exercise above, we're tasked with simplifying the expression \( \frac{12ab}{2ab^2} \). To simplify, follow these steps:

• First, break down each part of the expression. Look at the numbers (coefficients) separately from the variables.
• Divide the coefficients. In this case, divide 12 by 2 to get 6.
• Next, handle each variable. If a variable appears in both the numerator and the denominator, see if you can cancel them by subtracting exponents.

This will result in a simpler expression-free of any common factors. Making expressions simpler helps in quickly understanding and solving math problems, essentially like untangling a complicated knot to find the string ends.

Exponents

Exponents signify how many times a number, known as a base, is multiplied by itself. In algebra, exponents appear frequently, especially when simplifying expressions.

• In our example, we have \( b^1 \) in the numerator and \( b^2 \) in the denominator. We simplify exponents when dividing like bases by subtracting the exponent in the denominator from the exponent in the numerator.
• This gives us \( b^{1-2} \), or \( b^{-1} \). This negative exponent shows us that our base \( b \) moves to the denominator as \( \frac{1}{b} \).

Using exponent rules to simplify can streamline expressions, providing a clearer path to solving more complex problems. Understanding exponents helps in reducing expressions efficiently, critical in algebraic problem-solving.

Algebraic Fractions

Algebraic fractions are expressions where the numerator and/or the denominator are algebraic expressions. They are similar to regular fractions but involve variables along with numbers. In simplifying algebraic fractions, the primary goal is to eliminate the fraction by 'canceling out'.

• First, rewrite the expression so that it can be easily simplified. This might involve changing division into multiplication by the reciprocal.
• Identify and cancel out common factors in the numerator and denominator. For \( \frac{12ab}{2ab^2} \), this involved canceling the variable \(a\), and reducing \(b\)'s powers.
• After canceling, rewrite the expression without a denominator, highlighting the simplified form, as \(6b^{-1}\) or \(\frac{6}{b}\).

Understanding algebraic fractions and how to simplify them is essential. This not only makes expressions easier to solve but also lays the foundation for more complex algebraic operations.