Question

Simplify. \(\frac{36}{45 a} \div \frac{-9 a}{5}\)

Short answer

The simplified form of the given complex fraction is \(\frac{-4}{9 a^2}\)

Step-by-step solution

Apply the Keep-Change-Flip Rule

We rewrite the division operation as a multiplication using the Keep-Change-Flip rule, thus changing \(\frac{36}{45 a} \div \frac{-9 a}{5}\) to \(\frac{36}{45 a} * \frac{5}{-9 a}\)

Multiply Numerators and Denominators Separately

We multiply the numerators together and the denominators together. This results in \(\frac{36 * 5}{45 a * -9 a}\).

Simplify

Simplifying the result, we get \(\frac{180}{-405 a^2}\). We can further simplify this as \(\frac{-4}{9 a^2}\), by dividing both the numerator and the denominator by the common factor, 45.

Key concepts

Keep-Change-Flip Rule

When you encounter a problem that asks you to divide one fraction by another, you can simplify the process by using the 'Keep-Change-Flip rule'. This rule makes dividing fractions as straightforward as multiplying them. Here's what to do step-by-step:

First, keep the first fraction exactly as it is. Next, change the division sign to a multiplication sign. Lastly, flip the second fraction by switching its numerator and denominator. After applying this rule, your original division problem will turn into a multiplication problem, which is generally easier to solve. For example, in the given exercise \(\frac{36}{45 a} \div \frac{-9 a}{5}\), applying the Keep-Change-Flip rule transforms the expression into \(\frac{36}{45 a} * \frac{5}{-9 a}\).

Multiplying Fractions

Multiplying fractions might seem tricky at first, but it's actually quite simple once you get the hang of it. To multiply two fractions together, you just need to multiply the numerators (the top numbers) with each other and do the same with the denominators (the bottom numbers).

For instance, to multiply \(\frac{36}{45 a}\) and \(\frac{5}{-9 a}\), you multiply 36 by 5 to get the new numerator and multiply 45a by -9a to get the new denominator, resulting in \(\frac{36 \times 5}{45 a \times -9 a} = \frac{180}{-405 a^2}\). It is essential to multiply across without simplifying first—this ensures you have a clear view of all factors before you start reducing the fraction to its simplest form.

Dividing Rational Expressions

Dividing rational expressions involves simplifying complex fractions into their simplest form. A rational expression is simply the quotient of two polynomials. When you divide them, you are essentially performing the division of fractions, as shown in the Keep-Change-Flip rule.

Once you have transformed the division into multiplication of the inverted divisor, you should look for common factors in both the numerator and denominator and reduce them to their lowest terms to simplify the expression. In the exercise example, after applying the rule and multiplying the fractions, we get \(\frac{180}{-405a^2}\). Both the numerator and the denominator have a common factor of 45, which we use to simplify the expression to \(\frac{-4}{9a^2}\). Always ensure all common factors are eliminated to get the simplest form of the expression, and be attentive to the signs, particularly if either or both of the original expressions are negative.