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Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers.$\frac{1}{1+\sqrt{3}}$

$\frac{-1+\sqrt{3}}{2}$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 7

Rationalizing a Denominator

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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Rationalize the denominato…

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how to simplify 1/1, plus the skirt of three. You need to rationalize the denominator. So in order to rationalize the denominator, you have to multiply by the conjugated of one plus the sward of three. The conjugal. It is one minus the square root of three. In your numerator, the one has to distribute to both terms in the numerator. So in this case, one times one is one and one times a skirt of three is the squirt of three. So our numerator is one minus the square root of three. In the denominator, you need to use an extended distributive property. And the one in the first expression needs to multiply by both terms. So that gives you one times One is one one times. Um, the squirt of three is squirt of three. Because the squirt of three is being subtracted, it is technically a negative squirt of three. Um and then the squirt of three in the first expression also needs to be distributed toe both terms in the 2nd 1 So the square root of three times one is a positive squirt of three, and the scores of three times the negative squirt of three is going to be a negative square root of nine when you simplify in combined like terms, Um, the squirt of three. The negatives were 23 plus a positive squirt of three. Um, are opposites. So that creates zero. So you have one minus the square root of nine. And this word of nine is 31 minus three is negative too. So you just created a rational number in the denominator. Okay, Another way to write this is to take the negative from the numerator and make both of the terms. Sorry. Take the negative from the denominator. And to create a positive too. And the denominator, both of my terms and my numerator would have to be the opposite. So the opposite of one minus the skirt of three is a negative one. Plus a positive squirt of three, all divided by a positive too

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