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Simplify 13 over 13 plus two π.
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In this question, weβre looking to divide a real number by a complex number.
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And so we recall that to divide by a complex number, we write our quotient of the fraction and then we multiply both the numerator and the denominator of that fraction by the conjugate of the denominator.
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Then we find the conjugate of a complex number by changing the sign of the imaginary part.
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So if we have a complex number π§, which is π plus ππ, the conjugate, which we denote as π§ bar or sometimes π§ star, is π minus ππ.
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Now, the complex number that weβre dividing by is 13 plus two π.
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So weβre going to multiply by the conjugate.
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And we find that by changing the sign of the two π.
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This means the conjugate is 13 minus two π.
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And we need to multiply both the numerator and the denominator by this number.
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Letβs begin with doing that on the denominator.
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Weβre going to multiply 13 by 13 first.
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Thatβs 169.
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Then we multiply 13 by negative two π.
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And we get negative 26π.
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Two π minus 13 is positive 26π.
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And finally, we multiply the last term in each expression.
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Two π times negative two π is negative four π squared.
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Next, we spot that negative 26π plus 26π is zero.
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But of course, we also know that π squared is equal to negative one.
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So this simplifies to 169 minus four times negative one.
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And thatβs 169 plus four, which is 173.
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And so we have the denominator of our fraction.
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The numerator is a little more straightforward.
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We simply take the 13 here and we multiply it by both parts of our complex number.
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13 times 13 is 169, and 13 times negative two π is negative 26π.
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Weβre going to represent this using the standard form of a complex number, π plus ππ.
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And when we do, we get 169 over 173 minus 26 over 173π.