Problem 40 - Algebra 2

Question

Use the fact that \(\frac{b}{c}=a^{a} \div \frac{c}{d}\) to simplify each rational expression. State any restrictions on the variables. $$ \frac{\frac{3 a^{3} b^{3}}{a-b}}{\frac{4 a b}{b-a}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( -\frac{3 a^{2} b^{2}}{4} \) and the restriction on the variables is that \(a \neq b\).

Step 1 - Recognise the reciprocal relationship

First, recognise that the denominator of the division, \( \frac{4 a b}{b-a} \), is the reciprocal of \( -\frac{4 a b}{a-b} \). The denominators are negations of each other, so one is the negation of the reciprocal of the other.

Step 2 - Rewrite the division as multiplication

Rewrite the division as multiplication by the reciprocal. So, the division \( \frac{\frac{3 a^{3} b^{3}}{a-b}}{\frac{4 a b}{b-a}} \) is the same as multiplication \( \frac{3 a^{3} b^{3}}{a-b} * -\frac{b-a}{4 a b} \).

Step 3 - Multiply the fractions

Multiply the fractions. When multiplying fractions, it is done by multiplying the numerators of the fractions to get the new numerator and multiplying the denominators of the fractions to get the new denominator. Here, we get \( -\frac{3 a^{2} b^{2} (b-a)}{4 (a-b)} \).

Step 4 - Simplify the expression

Simplify the expression by cancelling out \( (b-a) \) in the numerator and denominator. The result is \( -\frac{3 a^{2} b^{2}}{4} \).

Step 5 - State the restrictions

State the restrictions on the variables. The denominator \(a - b\) in the original expression must not be equal to zero, so we have the restriction \(a \neq b\).

Key Concepts

Reciprocal

When working with rational expressions, understanding reciprocal relationships is a key concept. A reciprocal of a fraction is essentially what you get by flipping the numerator and the denominator. For example, the reciprocal of \( \frac{4ab}{b-a} \) is \( \frac{b-a}{4ab} \). This reciprocal flip is important when solving division problems involving fractions.
The idea behind the reciprocal is simple: it helps turn division into multiplication, which is usually easier to handle. So, if you are dividing by a fraction, you can multiply by its reciprocal instead. This method preserves the value of the original expression while simplifying calculations.

• Remember: If you multiply a fraction by its reciprocal, you get 1 (as in \( \frac{b-a}{b-a} \)).
• Using reciprocals is a useful strategy to simplify complex rational expressions.

Division of Fractions

Dividing fractions might seem tricky at first, but once you know the trick, it's quite straightforward. Instead of directly dividing, you can multiply by the reciprocal of the divisor fraction. For instance, if you have \( \frac{x}{y} \div \frac{w}{z} \), you can change this into \( \frac{x}{y} \times \frac{z}{w} \).
Once you rewrite the division problem as a multiplication problem, you can then multiply the numerators together and the denominators together. This results in a simplified expression that is often much easier to understand and simplify.

• Always remember to change division into multiplication by using the reciprocal of the second fraction.
• This technique helps maintain the equality of the equation, while making it simpler to solve.

Variable Restrictions

When dealing with rational expressions, it's important to be aware of variable restrictions. These are values that the variables cannot take in order to avoid undefined expressions.
For example, in the expression \( \frac{3 a^3 b^3}{a-b} \), \( a-b \) is in the denominator. This means \( a \) should never equal \( b \) because dividing by zero is undefined.
Identifying restrictions helps ensure that the expression is valid and that operations do not lead to mathematical errors. Always:

• Check each denominator to ensure it does not equal zero.
• Consider all parts of the expression where division might occur and assess possible zero values.

By keeping track of these variable restrictions, you'll ensure your solutions are accurate and mathematically sound.