Problem 99 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{27 p^{4}}{35 q} \div \frac{9 p}{21 q}$$

Step-by-Step Solution

Answer

The simplified result is \(\frac{9 p^{3}}{5}\).

Step 1: Rewrite the Division as Multiplication

The first step in simplifying a division of fractions is to rewrite the expression as a multiplication by the reciprocal of the divisor. Given the problem \(\frac{27 p^{4}}{35 q} \div \frac{9 p}{21 q}\), we rewrite it as follows: \(\frac{27 p^{4}}{35 q} \times \frac{21 q}{9 p}\).

Step 2: Multiply the Numerators and Denominators

Now multiply the numerators and the denominators across the fractions. This gives:\[\frac{27 p^{4} \times 21 q}{35 q \times 9 p}\] which can be rewritten as:\[\frac{27 \times 21 \times p^{4} \times q}{35 \times 9 \times q \times p}\]

Step 3: Cancel Common Factors

Identify and cancel the common factors in the numerator and the denominator. We have common factors of 9, \(q\), and \(p\): \[\frac{(9 \times 3) \times 21 \times p^{4} \times q}{35 \times (9) \times q \times p}\]Cancel out the \(9\), \(q\), and \(p\):\[\frac{3 \times 21 \times p^{3}}{35}\]

Step 4: Simplify the Expression

Simplify the remaining expression. Multiply the numbers in the numerator:\(3 \times 21 = 63\), and we have:\[\frac{63 p^{3}}{35}\]The greatest common divisor of 63 and 35 is 7. So divide both by 7:\[\frac{63 \div 7 \times p^{3}}{35 \div 7} = \frac{9 p^{3}}{5}\]

Key Concepts

Division of Rational Expressions

When confronted with dividing rational expressions, the process can be simplified by transforming the division into a multiplication problem. The key step is to multiply by the reciprocal of the divisor.

Consider the expression: \(\frac{27 p^{4}}{35 q} \div \frac{9 p}{21 q}\). Instead of dividing directly, we convert the operation into:\

• Multiplying by the reciprocal of the second fraction, which is \(\frac{21 q}{9 p}\).

This converts the original problem into a multiplication exercise: \(\frac{27 p^{4}}{35 q} \times \frac{21 q}{9 p}\).

By reversing the division into a multiplication, the operation becomes more straightforward, allowing for further steps of simplification and eventual cancellation of similar factors.

Simplifying Expressions

Simplifying expressions involves transforming complicated fractions into their simplest forms. This process helps in making the math more understandable and manageable.

For the expression obtained after rewriting as multiplication, \(\frac{27 p^{4} \times 21 q}{35 q \times 9 p}\), we simplify by:

• First, multiplying the numerators and denominators separately.
• Then, spotting common factors in both the numerator and the denominator.

In our example:
- The numerator can be broken down into specific factors: \(27 \times 21 \times p^{4} \times q\).
- The denominator is \(35 \times 9 \times q \times p\).

By identifying common factors such as \(q\), \(p\), and \(9\), these elements can be canceled out, leading to a much simpler expression: \(\frac{3 \times 21 \times p^{3}}{35}\). This process illustrates how rational expressions can be made less cumbersome through simplification.

Multiplication of Fractions

Multiplying fractions is an essential skill when handling rational expressions. It's straightforward with the crucial task of multiplying across both the numerators and the denominators.

Using our expression \(\frac{27 p^{4}}{35 q} \times \frac{21 q}{9 p}\), here’s how the process works:

• Multiply the numerators: \(27 \times 21 \times p^{4} \times q\).
• Do the same with the denominators: \(35 \times 9 \times q \times p\).

Once fractions are multiplied, the next step is to simplify the result further, which entails reducing by canceling out matching factors:

In our instance, identifying \(q\), \(9\), and \(p\) as common factors allows us to streamline the expression. With the remaining components, further simplification requires calculating \(3 \times 21\) to yield \(63\), and recognizing the greatest common divisor. Hence:\

• Divide \(63 \text{ by } 7\), and \(35 \text{ by } 7\) to obtain the final reduced form, \(\frac{9 p^{3}}{5}\).

This method of multiplication and followed simplification ensures the most reduced, clear form of a rational expression, making subsequent calculations easier.