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Put negative 18 minus nine π over three π in the form π plus ππ.
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In order to write our fractional expression in the form π plus ππ, we essentially need to find a way to get rid of this imaginary number on the denominator.
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And so there are two things that we could do.
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Letβs consider both methods.
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The first method involves rewriting our fraction somewhat.
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By reversing the process for adding fractions, we can write it as negative 18 over three π minus nine π over three π.
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And this is really useful because π divided by itself will be one.
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So this second fraction becomes nine divided by three, which is of course equal to three.
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So this simplifies a little bit to negative 18 over three π minus three.
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Negative three then is the π part of our expression.
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But what do we do with negative 18 over three π?
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Well, we can simplify it a little bit.
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18 divided by three is six and three divided by three is one.
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So it simplifies further to negative six over π minus three.
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We need to find a way to remove this π from the denominator of our fraction.
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And so we go back to the definition of π.
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π is the number such that π squared is equal to negative one.
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So if we can square the denominator, weβll end up with a simple integer that we can work with.
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Of course, we cannot just multiply the denominator of our expression.
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We need to do the same to the numerator.
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This is the equivalent then of multiplying by one.
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And so it doesnβt change the value of our fraction.
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We get then negative six π over π squared minus three.
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But of course, we know that π squared is equal to negative one.
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So we get negative six π over negative one minus three.
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And since negative six divided by negative one is six, this becomes six π minus three.
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We want to write the real part of our complex number first.
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So we just switch these numbers around.
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And we find that negative 18 minus nine π over three π is negative three plus six π.
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Now there is an alternative way to answer this.
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And that is by multiplying both the numerator and the denominator by π straightaway.
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When we do this, we get three π squared as the denominator of our fraction.
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And the numerator is π times negative 18 minus nine π.
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We distribute that π across the parentheses on the numerator of our fraction to get negative 18π minus nine π squared.
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Then we replace the π squared on the denominator with negative one.
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And so the denominator of our fraction becomes three times negative one which is just negative three.
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We then replace the π squared on our numerator with negative one.
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And so we get negative 18π minus nine times negative one which is negative 18π plus nine.
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We now see that we can divide both parts of our numerator by that negative three.
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Negative 18π divided by negative three is positive six π, and nine divided by negative three is negative three.
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Once again, weβve shown that writing our number in the form π plus ππ and we get negative three plus six π.