Rationalize the denominator. −527-\frac{5}{2\sqrt7}

− 5 7 14 -\frac{5\sqrt7}{14} − 14 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">

Rationalize the denominator. − 5 2 7 -\frac{5}{2\sqrt7}

Hey, everyone. So let's take a look at this practice problem. I wanna rationalize the denominator, and I've got negative 5 on top of 2 square root 7. Remember, leaving fractions or sorry, leaving radicals inside of the bottoms of fractions and denominators is bad. You can't do this. Every time you see this, a little alarm should go off in your head. So how do we do this? We rationalize it. Basically, what we're gonna do is we're gonna have to multiply by something on the top and bottom to get rid of this square root here that's on the bottom. And usually what we do is we multiply by whatever is on the bottom, right? So can we just multiply by 2 square root of 7 and 2 square root of 7? You totally could, but actually, you're just gonna create more work for yourself if you do this. The best thing to do here is only just multiply by whatever is inside of the radical. So you could just multiply by square root of 7 over square root of 7, and you're gonna be perfectly fine. Alright? So how do I do this? Well, what's gonna happen is on the bottom, if you have 2 times the square root of 7 times the square root of 7, then basically this just turns out to be a perfect square. And if you square roots, if you just combine 2 things or if you multiply 2 squares that are the same, you just drop the square root and this just becomes 2 times 7. On the top, what happens is this just becomes negative 5 times radical 7. Now we can do here is we're not quite done. We've gotten rid of the radical on the bottom. We could just multiply these two numbers, and that's pretty straightforward. This is gonna be negative 5 square root of 7 divided by, and this is 14. Alright. So again, you could have multiplied by 2 square root of 7 and 2 square root of 7. You totally could have done that, but you you would have ended up with bigger numbers over here in the numerator and denominator that you could still further simplify. Alright. So that's really all there is to it. Let me know if you have any questions. Thanks for watching.

Rationalize the denominator. − 5 2 7 -\frac{5}{2\sqrt7}

− 5 7 14 -\frac{5\sqrt7}{14} − 14 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">

− 5 14 -\frac{5}{14} − 14 5

− 5 7 2 -\frac{5\sqrt7}{2} − 2 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">

− 10 7 14 -\frac{10\sqrt7}{14} − 14 10 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">

11. Roots, Radicals, and Complex Numbers

Multiplying, Dividing, and Rationalizing Radicals

Beginning & Intermediate Algebra

11. Roots, Radicals, and Complex Numbers

Multiplying, Dividing, and Rationalizing Radicals

Struggling with Beginning & Intermediate Algebra?

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Multiple Choice

Rationalize the denominator.
$- \frac{5}{2 \sqrt{7}}$

$-\frac{5}{14}$

$-\frac{5\sqrt7}{2}$

$-\frac{5\sqrt7}{14}$

$-\frac{10\sqrt7}{14}$

Verified step by step guidance

Identify the expression to rationalize: $-\frac{5}{2\sqrt{7}}$. The denominator contains a square root, which we want to eliminate.

Multiply both the numerator and the denominator by $\sqrt{7}$ to rationalize the denominator. This uses the property that multiplying by $\frac{\sqrt{7}}{\sqrt{7}}$ is equivalent to multiplying by 1, so the value of the expression does not change.

After multiplication, rewrite the expression as $-\frac{5 \times \sqrt{7}}{2 \times \sqrt{7} \times \sqrt{7}}$.

Simplify the denominator using the fact that $\sqrt{7} \times \sqrt{7} = 7$, so the denominator becomes \$2 \times 7 = 14$. The expression now looks like $-\frac{5\sqrt{7}}{14}$.

Check if the fraction can be simplified further by looking for common factors in the numerator and denominator. If none exist, the expression is fully rationalized.

2:58m

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