Simplify the following complex fraction. [tex]$\frac{\frac{z+6}{z-4}}{\frac{z^2-36}{z^2-16}}$[/tex]
Answer (2)
The complex fraction z 2 − 16 z 2 − 36 z − 4 z + 6 simplifies to z − 6 z + 4 by rewriting it as a division problem, multiplying by the reciprocal, factoring the quadratic expressions, and canceling common factors.
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Rewrite the complex fraction as a division problem.
Rewrite the division as multiplication by the reciprocal.
Factor the quadratic expressions.
Cancel common factors to obtain the simplified expression: z − 6 z + 4
Explanation
Understanding the Problem We are asked to simplify the complex fraction z 2 − 16 z 2 − 36 z − 4 z + 6 . This involves dividing one fraction by another, which is the same as multiplying by the reciprocal. We will factor the polynomials and cancel common factors to simplify the expression.
Rewriting as Division First, we rewrite the complex fraction as a division problem: z 2 − 16 z 2 − 36 z − 4 z + 6 = z − 4 z + 6 ÷ z 2 − 16 z 2 − 36
Multiplying by the Reciprocal Next, we rewrite the division as multiplication by the reciprocal: z − 4 z + 6 ÷ z 2 − 16 z 2 − 36 = z − 4 z + 6 ⋅ z 2 − 36 z 2 − 16
Factoring Quadratic Expressions Now, we factor the quadratic expressions: z 2 − 16 = ( z − 4 ) ( z + 4 ) z 2 − 36 = ( z − 6 ) ( z + 6 )
Substituting Factored Expressions Substitute the factored expressions into the expression: z − 4 z + 6 ⋅ z 2 − 36 z 2 − 16 = z − 4 z + 6 ⋅ ( z − 6 ) ( z + 6 ) ( z − 4 ) ( z + 4 )
Canceling Common Factors Cancel common factors: z − 4 z + 6 ⋅ ( z − 6 ) ( z + 6 ) ( z − 4 ) ( z + 4 ) = ( z − 4 ) ( z − 6 ) ( z + 6 ) ( z + 6 ) ( z − 4 ) ( z + 4 ) We can cancel the common factors ( z + 6 ) and ( z − 4 ) from the numerator and the denominator.
Simplifying the Expression Simplify the expression by canceling the common factors: ( z − 4 ) ( z − 6 ) ( z + 6 ) ( z + 6 ) ( z − 4 ) ( z + 4 ) = z − 6 z + 4
Final Answer Therefore, the simplified expression is z − 6 z + 4 .
Examples
Complex fractions might seem abstract, but they appear in various real-world scenarios. For instance, when calculating the average speed of a journey with varying speeds over different segments, you might encounter a complex fraction. Similarly, in electrical engineering, when dealing with parallel circuits and impedances, complex fractions can arise when simplifying the total impedance. Understanding how to simplify these fractions helps in making these calculations more manageable and interpretable.
