[tex]\frac{1}{w^2-9} \div \frac{w+3}{w-3}[/tex]
Answer (2)
Rewrite the division as multiplication by the reciprocal: w 2 − 9 1 × w + 3 w − 3 .
Factor the denominator: w 2 − 9 = ( w − 3 ) ( w + 3 ) .
Substitute the factored form: ( w − 3 ) ( w + 3 ) 1 × w + 3 w − 3 .
Cancel the common factor ( w − 3 ) and simplify: ( w + 3 ) 2 1 .
The simplified expression is ( w + 3 ) 2 1 .
Explanation
Understanding the Problem We are asked to simplify the expression w 2 − 9 1 ÷ w − 3 w + 3 . This involves dividing one rational expression by another. To do this, we will rewrite the division as multiplication by the reciprocal of the second fraction. Then, we will factor the denominator and simplify by canceling common factors.
Rewriting the Division First, we rewrite the division as multiplication by the reciprocal: w 2 − 9 1 ÷ w − 3 w + 3 = w 2 − 9 1 × w + 3 w − 3
Factoring the Denominator Next, we factor the denominator w 2 − 9 as a difference of squares: w 2 − 9 = ( w − 3 ) ( w + 3 )
Substituting the Factored Form Now, we substitute the factored form into the expression: ( w − 3 ) ( w + 3 ) 1 × w + 3 w − 3
Canceling Common Factors We can cancel the common factor ( w − 3 ) from the numerator and denominator: ( w − 3 ) ( w + 3 ) 1 × w + 3 w − 3 = ( w + 3 ) ( w + 3 ) 1
Simplifying the Expression Finally, we simplify the expression: ( w + 3 ) ( w + 3 ) 1 = ( w + 3 ) 2 1 Thus, the simplified expression is ( w + 3 ) 2 1 .
Examples
Rational expressions are useful in many areas of science and engineering. For example, when designing electrical circuits, you might use rational expressions to describe the relationship between voltage, current, and resistance. Simplifying these expressions can help you understand the circuit's behavior and optimize its performance. Also, in physics, rational expressions can appear when dealing with lenses and optics, where they help describe how light bends and focuses.
To simplify the expression w 2 − 9 1 ÷ w − 3 w + 3 , we rewrite it as multiplication by the reciprocal, factor the denominator, cancel common factors, and arrive at the simplified result of ( w + 3 ) 2 1 .
;
