Rationalize the denominator and simplify the radical expression.75−6\frac{\sqrt7}{5-\sqrt6}

5 7 + 42 19 \frac{5\sqrt7+\sqrt{42}}{19} 19 5 7 <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"> + 42 <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. 7 5 − 6 \frac{\sqrt7}{5-\sqrt6}

Welcome back everyone. So in this problem, I've got the square root of 7 divided by 5 minus the square root of 6. So I've got a square root in the bottom of a fraction, and remember that's bad. So I'm gonna have to rationalize this. So let's get started here. Now remember, I wanna multiply the top and the bottom by something to get rid of the square roots. And because I don't have a single term in here, I can't just multiply by the square root of 6. If I try to do this, remember that the square root is gonna distribute into the 5, and I'm still gonna have a square root over here. So what do we do? We have to multiply by the conjugates. So remember that the conjugate is basically just the same exact expression, but you sort of flip the sign that's between the two terms. So the conjugate from here is gonna be 5+q That's what I have to multiply on the top and the bottom. So it's not square root of 6. It's 5 plus the square root of 6. Remember, whatever you multiply on the bottom, you have to also multiply on the top. Otherwise, you're changing the value of that equation here. So let's go ahead and get started here. Now if you think about this, what happens here is I have, I had the same terms. Right? I have 5 minus the square root of 6 and 5 plus the square root of 6. The only difference is as as a minus sign and a plus sign. Now this actually kind of looks like our special product formula, the difference of squares. So it's kind of looked like a plus b, a minus b, and so we know the answer is gonna be a squared minus b squared. So that's what we can do here. This is kind of like my a and this is kind of like my b over here. So that means that what our answer should be is on the bottom, we should have 5 squared, which is just 5 5 squared minus the square roots of 6, that's my b term, and and that's gonna be squared as well. Now what happens on the top? Well, I'm basically just gonna distribute this square root of 7 into both terms. And then here, I'm gonna get 5 square root of 7 plus, and then I've got square root of 6 times the square root of 7. Alright. And we can simplify that further. So what happens here? Well, I've got the 5 square root of 7 that stays the same, And then what happens here is I have 2 square roots that are multiplied together. And remember, what happens is when you have multiplied square roots, you can basically just sort of combine them into 1 large square root, and this is really just gonna be the square root of 42. Alright? And then on the bottom, what happens is I just get 5 squared, which is 25, minus the square root of 6 squared. So I'm taking 6, and I'm squaring it, but then I'm taking the square root of it. So I'm basically just sort of undoing it, and this really just becomes 6. Right? The square root of 6 squared just becomes 6. Alright. So if we simplify this even further, what happens is can we actually simplify the square root of 42? Well, no, because we saw that this really just goes down to 67, and both those can't be simplified any further. So it's actually perfectly fine to leave this as the square root of 42 over here. And on the bottom, what you get is 25 minus 6, which really just becomes 19. And so now I have square roots on the top, which is perfectly fine, and I have no square roots on the bottom. And so this expression over here is simplified, and I rationalize the denominator. Alright, folks. That's it for this one. Thanks for watching.

Rationalize the denominator and simplify the radical expression. 7 5 − 6 \frac{\sqrt7}{5-\sqrt6}

5 7 + 42 19 \frac{5\sqrt7+\sqrt{42}}{19} 19 5 7 <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"> + 42 <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 19 \frac{\sqrt7}{19} 19 7 <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">

5 7 + 42 − 11 \frac{5\sqrt7+\sqrt{42}}{-11} − 11 5 7 <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"> + 42 <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">

5 7 + 42 21 \frac{5\sqrt7+\sqrt{42}}{21} 21 5 7 <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"> + 42 <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">

9. Roots, Radicals, and Complex Numbers

Multiplying, Dividing, and Rationalizing Radicals

Intermediate Algebra

9. Roots, Radicals, and Complex Numbers

Multiplying, Dividing, and Rationalizing Radicals

Struggling with Intermediate Algebra?

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Multiple Choice

Rationalize the denominator and simplify the radical expression.
$\frac{\sqrt{7}}{5 - \sqrt{6}}$

$\frac{\sqrt7}{19}$

$\frac{5\sqrt7+\sqrt{42}}{-11}$

$\frac{5\sqrt7+\sqrt{42}}{21}$

$\frac{5\sqrt7+\sqrt{42}}{19}$

Verified step by step guidance

Identify the expression to rationalize: $\frac{\sqrt7}{5-\sqrt6}$. The denominator contains a radical, so we want to eliminate it by multiplying by the conjugate.

Write the conjugate of the denominator: \$5 + \sqrt6$. Multiply both numerator and denominator of the fraction by this conjugate to keep the value equivalent.

Multiply the numerators: $\sqrt7 \times (5 + \sqrt6) = 5\sqrt7 + \sqrt{42}$. Multiply the denominators using the difference of squares formula: $(5 - \sqrt6)(5 + \sqrt6) = 5^2 - (\sqrt6)^2 = 25 - 6$.

Simplify the denominator: \$25 - 6 = 19$. So the expression becomes $\frac{5\sqrt7 + \sqrt{42}}{19}$.

Check if the numerator or denominator can be simplified further. Since 19 is prime and the radicals cannot be simplified further, the expression is now rationalized and simplified.

2:58m

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