Solving a tricky SAT square root problem

How do we combine square root numbers? Here we will work out the square root of 20 minus the square root of 5, i.e. sqrt(20)-sqrt(5)=? You need to know how to simplify square root numbers and how to combine square root numbers for your algebra class! Subscribe to @bprpmathbasics for more algebra tutorials.

0:00 sqrt(20)-sqrt(5)=?
4:21 You try sqrt(18)-sqrt(8)=?
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#math #algebra #mathbasics

[asheep]

Thought Process:
sqrt(20) = sqrt(5 x 4)
sqrt(5 x 4) = 2sqrt(5)
2sqrt(5) - sqrt(5) = sqrt(5)

[rattanversha]

(√18)-(√8) is
(√9*√2)-(√4*√2) furthermore
3(√2)-2(√2)
Radicals are same so we can subtract it, right
So 3-2=1 Then,
1(√2) or simply (√2)

[royalredbird]

Huh! It was easy, I solved it right away!!

[theeternalswrd]

You can also do it this way. If you notice that the radicands multiply to a perfect square, solve it algebraically by squaring both sides. So initially, the problem is sqrt(20) - sqrt(5) = x. When squaring both sides, you end up with sqrt(20)^2 - 2*sqrt(20*5) + (- sqrt(5))^2 = x^2. This yields 20 - 20 + 5 = x^2 or 5 = x^2. We know the solution is positive since sqrt(20) - sqrt(5) is positive, so we can ignore the negative solution to this quadratic, so the solution is x = sqrt(5).

[yazziiiieee]

This man explained it so well, but it didn't have to be so lengthy. This could've been done by prime factorization and grouping the factors into groups of two.

[TakeaSipBabes]

This is a nice trick to know. Back in school I used to just put everything into a bracket and multiply it with itself so it would be something like (a-b)^2 which equals a^2 - 2ab +b^2 and then apply square root to this result; this way you get rid of the square root and ab would always give you a perfect square. If ab is not a perfect square this technique wouldnt work but neither would his.

[wongkitty]

My approach:
Let the whole equation be x, and square both sides
then we have (sqrt(20) - sqrt(5))^2 = x^2
20 - 2 * sqrt(20) * sqrt(5) + 5 = x^2
20 - 2 * sqrt(20 * 5) + 5 = x^2
20 - 2*10 + 5 = x^2
5 = x^2
x = sqrt(5)
That's it

[ragerasse]

Thought process:

2 is the square root of 4, so the square root of 5 is close to 2.
4 is the square root of 16 but since that is a bit further away from 20, changing the number to 4.5 is a bit over 20 so that works.

4.5 - 2 =2.5

Now since the answer asks for a square root, that number must be squared.

2.5 × 2.5 is around and closest to 5.

Final answer: square root 5.

[HalifaxHercules]

For the tryout question, the answer is "D", square root of 2.

Square root of 18 is the same as 3 root 2, and square root of 8 is 2 root 2.

3 root 2 - 2 root 2 gives you root 2.

[nabeelmohideen]

As an Indian 9th grader this was not that tricky rather I think it's one of the places where i can easily score good marks

[kang]

thought process:

A = sqrt(20) - sqrt(5)
A^2 = (sqrt(20) - sqrt(5))^2
We can use a^2-2ab^-b^2
sqrt(20)^2 - 2 * sqrt(20 x 5) + sqrt(5)^2
20 - 2 * 10 + 5
A^2 = 5
sqrt(A^2)= sqrt(5)

mine seem to be different from others

[lukewalker]

The other way to do it without recalling the mathmatical relationhship of breaking out a square root under the radical is:
Simply estimate the square root of 20 which is between 4 and 5...let's call it 4.5.
Then do basic square root of 5 in your head and say that is ~ 2.2
Perform 4.5-2.2 = 2.2 or 2.3 and square it...again in your head = 5
So answer is square root of 5. This is made possible by multiple choice doing basic arithmetic in your head aka deduction.
Of course, the mathematical answer is pretty well known as well for people that passed math class and I have had a lot of calculus so not a big deal...lol.

[xtz]

Another thought process

In this case, we can think with bounding the values.
Now that the perfect square roots are written down, you can evaluate that 4 < sqrt(20) < 5, and 2 < sqrt(5) < 3, so the answer is roughly 2, which in this case only corresponds with sqrt(5).
In tests like SAT, where you are given answers to choose from, and not specific values, this can be nice to use.

This can also help in general when you need to estimate the value of a square root, like sqrt(55) would be between 7 and 8.

Edit: for the bonus question, sqrt(18) is slightly above 4, and sqrt(8) is slightly below 3, so the answer would be slightly above 1, which in this case is ~1.4 = sqrt(2)

[darkshadeyt]

Sqrt(18) - Sqrt(8)
(Multiply everything by root 2)
=> {Sqrt(36) - Sqrt(16)}/(Sqrt(2)
= (6-4)/Sqrt(2)
= 2/Sqrt(2)
=> Sqrt(2)
This method is satisfying

[moefinesse]

Sqrt18-sqrt8
Sqrt (9*2) - sqrt (4*2)
3sqrt 2 - 2sqrt 2
Sqrt 2
Answer D

[tom-kzpb]

This is the kind of problem that seems tricky when you first glance at it, but is trivial when you think about it for a few seconds. It is ironic how things can be that way, that what you think is tricky is really simple, or conversely, what you think is simple is really quite tricky.

[mrutyunjaymallik]

My method :-
We begin by taking the square of the expression.
(√20 - √5)^2
= 20 + 5 - 2 ( √100) [ By using the formula = ( a - b)^2 = a^2 + b^2 - 2ab
=20 +5 - 2(10)
=25-20
=5
= If, (√20 - √5)^2 = 5, then
= √20 - √5 = √5

[sriprasadjoshi]

Calculated before seeing video and got it right...

[theendofit]

I just did a bunch of aproximations and got it wright from that
Edit to explain. I knew that square root of 20 would be a little under 4.5 (after checking work later its 4.472)
And i knew square root of 5 is under 2.5 but i knew it was more under 2.5 then the other number was under 4.5 so wighted it by making it 2.3. 4.5-2.3=2.2 thus logically square root of 5 was the closest.

[vivekchoudhary]

Easiest questions i have seen in years 🙂