Rationalize the denominator and simplify the radical expression.2−32+3\frac{2-\sqrt3}{2+\sqrt3}2+32−3

7 − 4 3 7-4\sqrt3 7 − 4 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

Rationalize the denominator and simplify the radical expression. 2 − 3 2 + 3 \frac{2-\sqrt3}{2+\sqrt3} 2 + 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> 2 − 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z">

Everyone. So in this practice problem here, I've got a 2 plus square root of 3 on the bottom of a fraction. That's bad because I can't have square roots on the bottom. I'm gonna have to rationalize that it's the dominator over here. So I've got 2 minus square root of 3 over 2 plus square root of 3. What do I have to multiply this thing by to get rid of, that square root? I have to multiply by the conjugates. So remember, conjugate is take this number of this expression over here and flip the sign between the two terms. So the conjugate of this is 2 minus square root of 3. Alright? So now what happens is I'm gonna take that expression, multiply 2 minuteus square root of 3, and I have to do this on the bottom and the top. So basically, what ends up happening is on the bottom, I flip the sign, it's 2+2-three, 2 minuteus square root of 3, and on the top, I'm basically just multiplying 2 minuteus 3 by itself. Alright? So here's what this works out to remember this kind of works out to like a difference of squares. So 2, just becomes 2 squared, and then I have minus, and then I have the square root of 3, and that becomes squared. So what happens here is this just really just becomes 4 minuteus 3 on the bottom. What happens on the top? Well, basically, I can just foil. So this, what the the 2 and the 2 just becomes 4. The 2 and the radical 3 becomes minus 2 radical 3, but then I just do that again. I have another 2 radical 3. And then over here, I have radical 3 times radical 3, and, really, this just becomes plus 3. Alright? So the radical sort of undo each other, and they both become positive. Alright? So what does this simplify down to? Well, I have 4 +3, which is 7, and then I have 2 radical three and two radical three, but they're both negative. So that really just becomes -four radical three. Alright? So remember having radicals on the top? Perfectly fine, but that's simplified as possible. Now the last thing I can do here is just simplify the denominator. This just becomes 7 minuteus radical 4 radical 3 divided by 1, and basically, if you're just dividing by 1, you can just actually leave this alone. You don't actually have to express it as a fraction. So it turns out that this whole entire thing over here actually really just becomes 7 minuteus 4 square root of 3. Alright? Hopefully, that made sense. Thanks for watching.

Rationalize the denominator and simplify the radical expression. 2 − 3 2 + 3 \frac{2-\sqrt3}{2+\sqrt3} 2 + 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> 2 − 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

7 − 4 3 7-4\sqrt3 7 − 4 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

1 7 + 4 3 \frac{1}{7+4\sqrt3} 7 + 4 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> 1

7 + 4 3 7+4\sqrt3 7 + 4 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

1 7 − 4 3 \frac{1}{7-4\sqrt3} 7 − 4 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> 1

0. Fundamental Concepts of Algebra

Rationalize Denominator

Precalculus

0. Fundamental Concepts of Algebra

Rationalize Denominator

Struggling with Precalculus?

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Multiple Choice

Rationalize the denominator and simplify the radical expression.
$\frac{2-\sqrt3}{2+\sqrt3}$

$7-4\sqrt3$

$\frac{1}{7+4\sqrt3}$

$7+4\sqrt3$

$\frac{1}{7-4\sqrt3}$

Verified step by step guidance

Identify the expression to be rationalized: $ \frac{2 - \sqrt{3}}{2 + \sqrt{3}} $. The goal is to eliminate the square root in the denominator.

Multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of $2 + \sqrt{3}$ is $2 - \sqrt{3}$.

Perform the multiplication: $ (2 - \sqrt{3})(2 - \sqrt{3}) $ for the numerator and $ (2 + \sqrt{3})(2 - \sqrt{3}) $ for the denominator.

Use the difference of squares formula: $(a + b)(a - b) = a^2 - b^2$. Apply this to the denominator: $(2)^2 - (\sqrt{3})^2 = 4 - 3 = 1$.

Simplify the expression: The denominator becomes 1, so the expression simplifies to $ (2 - \sqrt{3})^2 $. Expand and simplify the numerator to get the final expression.

2:58m

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