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Chapter 7: Problem 80

Simplify each rational expression. $$\frac{16-y^{2}}{y(y-8)+16}$$

Short Answer

Expert verified
The simplified form of the given rational expression is \(-\frac{(y+4)}{y-4}\).

Step by step solution

01

Recognize the Difference of Squares on the Numerator

The numerator, \(16-y^{2}\), is a difference of squares. This type of expression can be factored into the product of the sums and differences of the two terms i.e., \((a^{2}-b^{2}) = (a+b)(a-b)\). So, \(16-y^{2}\) can be factored as \((4+y)(4-y)\).
02

Simplify the Denominator

The denominator, \(y(y-8)+16\), can be simplified by factoring. Distribute y into the brackets, i.e, \(y^2 - 8y + 16\). Notice that this can be factored into a perfect square \((y-4)^2\).
03

Simplify the Rational Expression

Now, let's simplify the rational expression: \(\frac{(4+y)(4-y)}{(y-4)^{2}}\). Note that \((4-y)\) and \((y-4)\) are the same, just that one is negative of the other. One \((y-4)\) from the denominator cancels out one of either \((4-y)\) or \((4+y)\) from the numerator, but this cancelation would carry a minus sign. Therefore, you get: \(-\frac{(4+y)}{(y-4)}\).
04

Final Simplification

Note that both the numerator and denominator have common factors, which causes confusion sometimes. So, let's rewrite \(4+y\) as \(y + 4\), yielding a final simplified expression of \(-\frac{(y+4)}{(y-4)}\).

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Most popular questions from this chapter

use the GRAPH or TABLE feature of a graphing utility to determine if the subtraction has been performed correctly. If the answer is wrong, correct it and then verify your correction using the graphing utility. $$\frac{x^{2}+4 x+3}{x+2}-\frac{5 x+9}{x+2}=x-2, x \neq-2$$

Describe how to identify the corresponding sides in similar triangles.

determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. I added \(\frac{5}{x-7}\) and \(\frac{3}{7-x}\) by first multiplying the second rational expression by \(-1\)

Two investments have interest rates that differ by \(1 \% .\) An investment for 1 year at the lower rate earns 175 . The same principal invested for a year at the higher rate earns 200 dollars . What are the two interest rates?

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-7}{y^{2}-16}+\frac{7-y}{16-y^{2}}$$
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