Rationalize the denominator.6+x−x\frac{6+\sqrt{x}}{-\sqrt{x}}

6 x − x − 1 \frac{6\sqrt{x}}{-x}-1 − x 6 x <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

Rationalize the denominator. 6 + x − x \frac{6+\sqrt{x}}{-\sqrt{x}}

everyone. So in this practice problem, I have a fraction with some square roots. I've got 6 +arelx or square root of x divided by negative square root of x. Remember, square roots on the bottoms of fractions, can't do it. I'm gonna have to rationalize the denominator. So what I have to do is I have to multiply this whole expression by something on the top and the bottom to get rid of it. Now again, usually, we just multiply by whatever is on the bottom, and in this case, you might be thinking, well, I just multiply by negative square root of x over negative square root of x, and that's perfectly fine. You can do that, but really you only have to just multiply by the radical itself. You only have to multiply by the square root of x. So if you do this, what happens is, on the bottom, you just get negative, and you get a square root of x times the square root of x. The square root drops, and all you just get is just x on the bottom, alright, plus the negative sign that was over here from the beginning. So then what happens to the top expression? You've got to basically distribute this square root of x inside of each of the terms because now you have addition over here. So it's kind of like a parenthesis. So what ends up happening here is if you distribute the square root of x into 6, you get 6 square root of x, and then over here when you get square root of x times square root of x, you just get x. Alright? So that's one way to do this and remember square roots on the numerator is perfectly fine, but I have no square roots in the denominators, and that's what I want. Now if you wanted actually to simplify this expression even further, what you could do is you could split out these fractions. We saw how to do that. So 6 times the square root of x divided by negative x plus and then you have x divided by negative x. And the reason I want to do this is because actually this simplifies down. This doesn't because I can't do anything with the square root of x and the x in the bottom, but this really just becomes 6 square root of x divided by negative x, and then really just is just x divided by negative x. So this just becomes negative one. Alright? So your professors may want you to write it in a form like this, where you're simplifying it the most you possibly could. But if you also just wrote it like this, again, it really just depends. If your professor wants something that's fully simplified or if you're if you're asked to fully simplify it, then you can't just leave it like this. You're gonna have to go all the way down here. But either of these are correct answers, and they and we've rationalized the denominators. Alright, folks. That's it for this one. Thanks for watching.

Rationalize the denominator. 6 + x − x \frac{6+\sqrt{x}}{-\sqrt{x}}

6 x − x − 1 \frac{6\sqrt{x}}{-x}-1 − x 6 x <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

6 x + x 6\sqrt{x}+x 6 x <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"> + x

6 x x + 1 \frac{6\sqrt{x}}{x}+1 x 6 x <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 x − x \frac{7\sqrt{x}}{-x} − x 7 x <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.
$\frac{6 + \sqrt{x}}{- \sqrt{x}}$

$\frac{6\sqrt{x}}{-x}-1$

$6\sqrt{x}+x$

$\frac{6\sqrt{x}}{x}+1$

$\frac{7\sqrt{x}}{-x}$

Verified step by step guidance

Identify the given expression: $\frac{6 + \sqrt{x}}{-\sqrt{x}}$. Our goal is to rationalize the denominator, which means eliminating the square root from the denominator.

Multiply both the numerator and the denominator by the conjugate or an expression that will remove the square root from the denominator. Since the denominator is $-\sqrt{x}$, multiply numerator and denominator by $-\sqrt{x}$ to rationalize.

Perform the multiplication in the numerator: $(6 + \sqrt{x})(-\sqrt{x}) = 6 \times (-\sqrt{x}) + \sqrt{x} \times (-\sqrt{x})$.

Simplify the terms in the numerator: \$6 \times (-\sqrt{x}) = -6\sqrt{x}$ and $\sqrt{x} \times (-\sqrt{x}) = -x$ because $\sqrt{x} \times \sqrt{x} = x$.

Simplify the denominator: $(-\sqrt{x}) \times (-\sqrt{x}) = (\sqrt{x})^2 = x$. Then write the new fraction with the simplified numerator and denominator.

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