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Simplifying Algebraic Expressions 2

<a href="https://www.prepswift.com/quizzes/quiz/prepswift-simplifying-algebraic-expressions-2" target="_blank">Simplifying Algebraic Expressions 2 Exercise</a>Sometimes we can simplify complex roots like this: $\sqrt{y^4-10y^2+25}$ We can use Identity 3 to recognize this is the same as $\sqrt{(y^2-5)^2}$ which is just $|y^2-5|$. We can only cancel the square root if we put the absolute value symbols ($|$ )around the result. Some complex fractions can be simplified too, but they take a bit of thinking: $\frac{2}{a^2-25}-\frac{1}{a^2+5a}$  We can start by factoring out the denominators using our identities: $\frac{2}{(a+5)(a-5)}-\frac{1}{a(a+5)}$ To get a common denominator (and cancel out what they don't share in common) we need to multiply the left by $\frac{a}{a}$ and the right by $\frac{a-5}{a-5}$. This will give both fractions a denominator of $a(a-5)(a+5)$: $\frac{2a}{a(a-5)(a+5)}-\frac{a-5}{a(a-5)(a+5)} = \frac{2a-a+5}{a(a-5)(a+5)} = \frac{a+5}{a(a-5)(a+5)}$ Finally, we can cancel out the $a+5$ to get: $\frac{1}{a(a-5)}$ Note that $a$ cannot equal $5$, $-5$, or $0$! <blockquote> Why does $-5$ not work? And indeed, it does when we look at the final expression $\frac{1}{a(a-5)}$. However, look at the original expression instead. Does $-5$ work in that case? No. And why? Because when you cancel out the $a + 5$, you're implicitly assuming that $a \neq -5$. Otherwise the domain of the expression will change - that is, values that could not work (such as $a = -5$) suddenly become allowed. And that's not OK - we simplify to make things easier for us, but the expression must functionally be the same thing (unless you know what you're doing - but that's not the subject of this discussion).  Another simple example is to consider the equation $x^2 = x$. A common mistake is to divide by $x$ on both sides, making the result $x = 1$. But notice that $x = 0$ works in the original equation! The reason is similar - when dividing by $x$, you're assuming that $x \neq 0$ when that need not be the case.  (We'll cover the concept of a domain in a later video/entry) </blockquote>