Problem 28 - Prealgebra

Question

Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$\frac{12 a^{2}}{5 b} \div \frac{6 a}{5 b^{2}}$$

Step-by-Step Solution

Answer

The quotient is \( 2ab \).

Step 1: Identify the Division

We start with the division problem given as \( \frac{12 a^{2}}{5 b} \div \frac{6 a}{5 b^{2}} \). We need to find the quotient of these two rational expressions.

Step 2: Reciprocal of the Divisor

To divide by a fraction, we multiply by its reciprocal. The reciprocal of the divisor \( \frac{6 a}{5 b^{2}} \) is \( \frac{5 b^{2}}{6 a} \).

Step 3: Rewrite as Multiplication

Now rewrite the original division problem as a multiplication. This becomes \( \frac{12 a^{2}}{5 b} \times \frac{5 b^{2}}{6 a} \).

Step 4: Multiply the Numerators

Multiply the numerators: \( 12 a^{2} \times 5 b^{2} = 60 a^{2} b^{2} \).

Step 5: Multiply the Denominators

Multiply the denominators: \( 5 b \times 6 a = 30 a b \).

Step 6: Simplify the Expression

Now we have the expression \( \frac{60 a^{2} b^{2}}{30 a b} \). Simplify by canceling common terms. The common factors of 60 and 30 reduce to 2, and \( a^{2} \) divided by \( a \) is \( a \), and \( b^{2} \) divided by \( b \) is \( b \) as well, leaving \( 2 a b \).

Key Concepts

Reciprocal

When dealing with division of fractions or rational expressions, the first key concept is the reciprocal. Let's say you have a fraction like \( \frac{3}{4} \). The reciprocal of this fraction is simply achieved by swapping the numerator and denominator, resulting in \( \frac{4}{3} \).
Knowing how to find the reciprocal is important because dividing by a fraction is the same as multiplying by its reciprocal. For example, if you want to divide \( \frac{12 a^{2}}{5 b} \) by \( \frac{6 a}{5 b^{2}} \), you would invert the second rational expression, yielding the reciprocal \( \frac{5 b^{2}}{6 a} \).
This process simplifies the division into something more manageable, essentially converting the operation into multiplication, which often feels more straightforward due to following standard multiplication rules and simplifications.

Multiplying Fractions

Once you have the reciprocal of the second fraction or rational expression, you can transform your division problem into multiplication. Recall that multiplying fractions - involves taking the product of the numerators to get the new numerator,- and the product of the denominators to get the new denominator.
In our example, after taking the reciprocal, you multiply: - The numerators: \( 12 a^{2} \times 5 b^{2} = 60 a^{2} b^{2} \).- The denominators: \( 5 b \times 6 a = 30 a b \).
This step is crucial because it sets the stage for the next step, simplifying the expression. The multiplied result often contains terms that need reducing for simplification, ensuring the final answer is as simple as possible, thus easy to interpret.

Simplifying Rational Expressions

After multiplying the fractions, you end up with a new rational expression that might seem complex at first. The goal now is to simplify this expression so that it is in its simplest form.
In the expression \( \frac{60 a^{2} b^{2}}{30 a b} \), notice that both the numerator and denominator have common factors. These need to be canceled out:

• The numbers 60 and 30 can both be divided by 30, simplifying to 2.
• The \( a^{2} \) in the numerator and \( a \) in the denominator simplify to just \( a \).
• The \( b^{2} \) in the numerator and \( b \) in the denominator simplify to \( b \).

After canceling these common terms, you're left with \( 2 a b \), which is the simplest form of the given rational expression. Simplifying always involves reducing the expression to its smallest terms by eliminating any factors common to both the numerator and the denominator, making calculations easier and clearer.