Rationalize the denominator. 6+x−x\frac{6+\sqrt{x}}{-\sqrt{x}}−x6+x

6x−x−1\frac{6\sqrt{x}}{-x}-1−x6x−1

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

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 to 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}}−x6+x

6x−x−1\frac{6\sqrt{x}}{-x}-1−x6x−1

6xx+1\frac{6\sqrt{x}}{x}+1x6x+1

7x−x\frac{7\sqrt{x}}{-x}−x7x

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0. Fundamental Concepts of Algebra3h 32m

Algebraic Expressions
37m

Exponents
32m

Polynomials Intro
19m

Multiplying Polynomials
36m

Factoring Polynomials
2m

Radical Expressions
15m

Simplifying Radical Expressions
31m

Rationalize Denominator
15m

Rational Exponents
4m

Pythagorean Theorem & Basics of Triangles
17m

1. Equations and Inequalities3h 27m

Linear Equations
31m

Rational Equations
21m

The Imaginary Unit
6m

Powers of i
11m

Complex Numbers
42m

Intro to Quadratic Equations
18m

The Square Root Property
11m

Completing the Square
12m

The Quadratic Formula
19m

Choosing a Method to Solve Quadratics
9m

Linear Inequalities
22m

2. Graphs1h 43m

Graphs and Coordinates
7m

Two-Variable Equations
23m

Lines
1h 12m

3. Functions & Graphs2h 17m

Intro to Functions & Their Graphs
35m

Common Functions
5m

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45m

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21m

Function Composition
29m

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Quadratic Functions
49m

Understanding Polynomial Functions
34m

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30m

5. Rational Functions1h 23m

Introduction to Rational Functions
9m

Asymptotes
35m

Graphing Rational Functions
38m

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Introduction to Exponential Functions
9m

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25m

The Number e
8m

Introduction to Logarithms
22m

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18m

Properties of Logarithms
27m

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35m

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10m

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45m

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26m

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23m

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14m

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9m

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11m

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6m

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32m

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14m

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10m

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21m

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28m

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44m

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28m

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1h 4m

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1h 3m

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16m

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9m

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49m

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30m

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19m

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28m

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32m

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23m

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27m

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22m

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11m

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29m

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26m

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29m

Graphing Other Common Polar Equations
39m

16. Parametric Equations1h 6m

Graphing Parametric Equations
12m

Eliminate the Parameter
32m

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21m

17. Graphing Complex Numbers1h 7m

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6m

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22m

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14m

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23m

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8m

Introduction to Matrices
1h 11m

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1h 3m

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42m

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Introduction to Conic Sections
6m

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26m

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33m

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36m

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40m

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12m

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33m

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25m

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15m

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11m

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46m

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47m

22. Limits & Continuity1h 49m

Introduction to Limits
41m

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40m

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27m

23. Intro to Derivatives & Area Under the Curve2h 9m

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44m

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18m

Limits of Sequences
9m

Area Under a Curve
56m

0. Fundamental Concepts of Algebra

Rationalize Denominator

Precalculus

0. Fundamental Concepts of Algebra

Rationalize Denominator

Problema precedenteProssimo problema

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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}$}{-x}$

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Problema precedenteProssimo problema

Guida verificata passo dopo passo

Identify the expression that needs rationalization: $ \frac{6 + \sqrt{x}}{-\sqrt{x}} $.

Multiply both the numerator and the denominator by the conjugate of the denominator, which is $ \sqrt{x} $, to eliminate the square root in the denominator.

The expression becomes $ \frac{(6 + \sqrt{x}) \cdot \sqrt{x}}{-\sqrt{x} \cdot \sqrt{x}} $.

Simplify the denominator: $ -\sqrt{x} \cdot \sqrt{x} = -x $.

Distribute $ \sqrt{x} $ in the numerator: $ 6\sqrt{x} + x $. The expression is now $ \frac{6\sqrt{x} + x}{-x} $.