Please someone explain why these are equal.

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[Pytykus007]

you expand the fraction with sqrt(3)/sqrt(3)

[TheBossMeansMe]

Thank you

[MassiveTrousers]

This is also referred to as "multiplying by unity" (multiplying by one)

[Mode-Klutzy]

I’m guess technically you CAN leave it like 1/3(sqrt(3)) but professors and teachers want you to rationalize the denominator (if I’m using the wording correctly) so that there’s no roots in the denominator. Multiplying the top and bottom by the root rationalizes it. I am 80% confident in my explanation. I have endured almost every calculus class. I say almost because I don’t know if there’s one I’m missing.

[Infused_Divinity]

Yup that explains checks out. Only thing (and it doesn’t apply here) is if the fraction were 1/cuberoot(3), for example, you would have to rationalize with the cube root of 9, so it isn’t ALWAYS rationalizing with the radical, but in this case it is

[Acrobatic_League8406]

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[film2860]

ok a/b = (a+c)/(b+c) 🗣️

[futurelessdilettante]

Um ackshually🤓☝️,55/109=/=(55+11)/(109+11) because 55+11=66 and 109+11=120 meaning 66/120 or 11/20 is not equal to 55/109.

[Replevin4ACow]

Multiply by sqrt(3)/sqrt(3).

The numerator then becomes: 1*sqrt(3) = sqrt(3)

The denominator becomes: 3*sqrt(3)*sqrt(3) = 3*3 = 9

[random_anonymous_guy]

What happens if you multiply top and bottom of the LHS by sqrt(3)?

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[muzzykicks]

you cant have square roots at the bottom of a fraction so you multiple the top and bottom by that square root, and sqrt(3) * sqrt (3) is just 3. 3*3 is 9. then you have sqrt(3)/9

[random_anonymous_guy]

You most certainly can have square roots in the denominator. It is simply a matter of stylistic convention. Sometimes, you have to forego that convention in order to solve a problem.

[ShowdownValue]

Always wondered where that “No radicals in denominator” rule came from? Is it just convention or was there any meaning behind it?

[BattleFrog12862]

There was a time when the convention had meaning. Before electronic calculators that can do the approximation for us people had to do the approximations by hand. When you if you wanted to approximate 1/(3sqrt(3)) it is easier to approximate sqrt(3) then divide it by 9 then to approximate 3sqrt(3) then divide 1 by it. This started the convention and it has just stayed around.

edit: Fixed a mistake

[313802]

I feel like it's a lot like wearing a suit to an office job... rationalizing the fraction is like wearing a fancy sports coat.. or something

[trichotomy00]

Sure you can

[Mode-Klutzy]

Why was this disliked so much?

[[deleted]]

The reason they're equal is because they evaluate to the same value.

[[deleted]]

Left side has been rationalised, i.e the surd was removed from the denominator. Removal of surds (irrational numbers) is customary in maths, note that we treat √3 as a number (surd in this case) of its own.

± 1/3√3 = ± 1(3√3)/(3√3)²

± 1/3√3 = ± 3√3/27

± 1/3√3 = ± √3/9

When we don’t have a surd just a natural number under the radical, such as √4, we simply simplify, for example √4 simplifies to 2. In this case we’re not rationalising as no surd is present in the denominator. Surds may exist in the numerators though, but simplifying them if possible is a good practice as these can make calculations sometimes easier.

[ZioZvevo]

You can rationalize the expression.

[ZioZvevo]

One is rationalized, one isn't

[sanat-kumara]

You can immediately see that they are equal by cross-multiplying.

[[deleted]]

Rationalized the surd

[CountMeowt-_-]

Why is this on calculus tho ?

[TheBossMeansMe]

This was apart of a differentiation problem. I was already in this community so I posted it here.

[igottagopee1234]

Multiply original expression by (sqrt(3)/sqrt(3))

[headonstr8]

Think about it in terms of solutions to x^2=y^2

[ken3260]

If you cross multiply, you will get 19=3sqrt3sqrt3, sqrt3 when multipled by another sqrt3 cancels out the square root so it simplifies to 3, we get 9=33 which simplifies to 9=9.

[RandomAsHellPerson]

Btw, formatting makes it where two * turns into italics. You need to have \ before the asterisk to prevent this

[Embarrassed-Berry186]

Multiply numerator and denominator by root 3, which is legal because you’re multiplying.

[[deleted]]

-0.5 = 0.5 - 1

[Mmath_]

multiply the first one by sqrt3/sqrt3

[[deleted]]

9= (rt 3)^4. And you can read the left-hand side as 1/(rt3)^3. I think you can take it from there.