Simplify the following rational expression: [tex]\frac{3 g^3 h^3}{g-h} \div \frac{4 g h}{h-g}[/tex]
Answer (2)
Rewrite the division as multiplication by the reciprocal.
Use the property that h − g = − ( g − h ) .
Cancel the common factor ( g − h ) .
Simplify the expression by cancelling common factors of g and h , resulting in − 4 3 g 2 h 2 .
Explanation
Understanding the problem We are asked to simplify the rational expression g − h 3 g 3 h 3 ÷ h − g 4 g h .
Rewriting the division To simplify the expression, we first rewrite the division as multiplication by the reciprocal: g − h 3 g 3 h 3 ÷ h − g 4 g h = g − h 3 g 3 h 3 ⋅ 4 g h h − g .
Using the property h-g = -(g-h) Next, we notice that h − g = − ( g − h ) . Substituting this into the expression, we get g − h 3 g 3 h 3 ⋅ 4 g h − ( g − h ) .
Cancelling (g-h) terms Now we can cancel the ( g − h ) terms: 1 3 g 3 h 3 ⋅ 4 g h − 1 .
Simplifying the expression Finally, we simplify the expression by cancelling common factors of g and h : 1 3 g 2 h 2 ⋅ 4 − 1 = − 4 3 g 2 h 2 .
Examples
Rational expressions are used in various fields, such as physics and engineering, to model relationships between different quantities. For example, in electrical engineering, rational functions can describe the impedance of a circuit as a function of frequency. Simplifying these expressions allows engineers to analyze and design circuits more efficiently. Similarly, in physics, rational expressions can appear in equations describing the motion of objects or the behavior of waves. Simplifying these expressions helps physicists to understand and predict the behavior of physical systems.
The expression g − h 3 g 3 h 3 ÷ h − g 4 g h simplifies to − 4 3 g 2 h 2 by rewriting the division as multiplication with the reciprocal, using a property of the subtraction, and then cancelling common factors.
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