Simplify the following rational expressions: [tex]$\frac{3 g^3 h^3}{g-h} \div \frac{4 g h}{h-g}$[/tex]
Answer (2)
Simplify the first expression by multiplying by the reciprocal and canceling common factors: g − h 3 g 3 h 3 ÷ h − g 4 g h = − 4 3 g 2 h 2 .
Simplify the second expression by factoring: 4 g − 4 h 3 g 3 h 3 − 3 g 3 h 2 = 4 ( g − h ) 3 g 3 h 2 ( h − 1 ) .
Simplify the third expression by recognizing ( g − h ) ( − g + h ) = − ( g − h ) 2 : ( g − h ) ( − g + h ) 12 g 4 h 4 = − ( g − h ) 2 12 g 4 h 4 .
The fourth expression is already simplified: − 4 3 g 2 h 2 .
The fifth expression is already simplified: 4 3 g 4 h 4 .
Explanation
Problem Analysis We are given five rational expressions to simplify. Our goal is to simplify each expression and present the simplified forms.
Simplifying the First Expression Let's simplify the first expression: g − h 3 g 3 h 3 ÷ h − g 4 g h . To divide by a fraction, we multiply by its reciprocal. Also, note that h − g = − ( g − h ) . Thus, we have: g − h 3 g 3 h 3 ÷ h − g 4 g h = g − h 3 g 3 h 3 ⋅ 4 g h h − g = g − h 3 g 3 h 3 ⋅ 4 g h − ( g − h ) = − 4 g h ( g − h ) 3 g 3 h 3 ( g − h ) . We can cancel out the ( g − h ) terms and simplify the fraction g h g 3 h 3 = g 2 h 2 . Therefore, the simplified expression is − 4 3 g 2 h 2 .
Simplifying the Second Expression Now, let's simplify the second expression: 4 g − 4 h 3 g 3 h 3 − 3 g 3 h 2 . We can factor out 3 g 3 h 2 from the numerator and 4 from the denominator: 4 g − 4 h 3 g 3 h 3 − 3 g 3 h 2 = 4 ( g − h ) 3 g 3 h 2 ( h − 1 ) . This expression is already in its simplest form.
Simplifying the Third Expression Next, we simplify the third expression: ( g − h ) ( − g + h ) 12 g 4 h 4 . Notice that − g + h = h − g , so we can rewrite the denominator as ( g − h ) ( h − g ) = − ( g − h ) 2 . Thus, we have: ( g − h ) ( − g + h ) 12 g 4 h 4 = − ( g − h ) 2 12 g 4 h 4 = − ( g − h ) 2 12 g 4 h 4 .
The Fourth Expression The fourth expression is already simplified: − 4 3 g 2 h 2 .
The Fifth Expression The fifth expression is also already simplified: 4 3 g 4 h 4 .
Final Simplified Expressions In summary, the simplified expressions are: g − h 3 g 3 h 3 ÷ h − g 4 g h = − 4 3 g 2 h 2 4 g − 4 h 3 g 3 h 3 − 3 g 3 h 2 = 4 ( g − h ) 3 g 3 h 2 ( h − 1 ) ( g − h ) ( − g + h ) 12 g 4 h 4 = − ( g − h ) 2 12 g 4 h 4 − 4 3 g 2 h 2 4 3 g 4 h 4
Examples
Rational expressions are useful in many areas of mathematics and physics. For example, in electrical engineering, they can be used to analyze circuits. If you have a circuit with resistors in parallel, the total resistance can be expressed as a rational expression. Simplifying these expressions can help engineers understand the behavior of the circuit and design it more effectively. Also, in computer graphics, rational expressions are used to represent curves and surfaces. Simplifying these expressions can help to optimize the rendering process and create more realistic images.
The simplified form of the expression g − h 3 g 3 h 3 ÷ h − g 4 g h is − 4 3 g 2 h 2 . This is achieved by multiplying by the reciprocal of the second fraction and canceling common terms. Careful treatment of the signs and exponents is essential during the simplification process.
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