How do you rationalize the denominator when simplifying a square root?

Question

I know how to simplify square roots, but I don't know what rationalizing the denominator means. I have two problems that I'm stuck on:

1) 3√7/12
I'm supposed to simplify this, but I don't know how 7/12 can be simplified any more than it is.

2) √128 + √50
Neither of these square roots can be simplified, but if you add them, you get √178, which I don't think can be simplified either. So is there anything you can do to simplify this problem?

Thanks!

Answer 1

1) u need to think: "is there any quadratic numbers in these
numbers?"
in the square root of 7/12:
- 7 isn't a quadratic number
- 12 is form of quadratic number => 12 = 4*3
so u still can simplify the 12 => 2sqrt3

then 3*sqrt(7/12) = 3*(1/2sqrt(7/3))
= 3/2* sqrt (7/3)

2) remember: u can't just add both of them, except they have
the same square root!!
ex: 5*sqrt5 + 10*sqrt5 = (5+10)*sqrt5 = 15sqrt5
in this case u can add them
in simple form: ab+cb= (a+c)b, isn't it?
so it's the same theory

sqrt128 + sqrt50 = sqrt(64*2) + sqrt(25*2)
= 8*sqrt2 + 5*sqrt2
= 13sqrt2

Answer 2

well i dunno about 1 because which is that? is that like... 3 x √7/12)???
if so.. √7/√12 you have to multiply the top and bottom by √12/√12 (it's just like multiplying by -- or doing nothing!)
then you get 3√84 all over 12... er... then i guess it's
√84)/4

you can simplify it more because 84=4x21 and 4 is a square.
then you do the square root of 4 and since 21 isn't a square, just leave it there.
√(4x21)/4= 2√21)/4 = √21)/2

ooer.

the second ones:
these can be simplified like..
128=2x64 and 50=2x25 so you then have under the radicals
√2x64) + √2x25)
64 and 25 are squares so do the operation as shown. You can take the square root of your 25 and your 64 to simplify and just leave the two under there.
8√2) + 5√2)
add them together just as if the √2 were variables and the 8 and 5 coefficients (they kinda are, aren't they, lol)
you get 13√2)

√128 + √50, however, is not √178. you can't just add them together like that.

Answer 3

3√7/12 = √7/4

√128 = √64 x √2 = 8√2
√50 = √25 x √2 = 5√2

√128 + √50 = 8√2 + 5√2 = 13√2

If you have to rationalize the denominator, you multiply both by the same radical

say 3/√5 = 3√5 / (√5*√5) = 3√5/5

Answer 4

Rationalize Square Roots

Answer 5

***
It seems to me that you do not understand how to simplify radicals. To do this, you must factor the radical and one of thos factors must be a perfect square. For example (i use v's for square root):
v12
The factors of 12 are 4 and 3. 4 is the perfect square.
v4 * v3
2v3
And thats simplified.

1) Rationalizing the denominator is usually required because the denominator of a fraction should NEVER be rooted. To solve this, multiply the numerator AND denominator by the root in the denominator. In this question:
3 * [v7(v12) / v12(v12)]
Simplify.
3 * [v7(v12)/12]
It doesnt look as simple, but it gets rid of the radical in the denominator. Then simplify the radicals.
3 * [2(v7)(v3) / 12]
This time, put the 3 in the numerator. Reduce 6 and 12.
(v7)(v3) / 2

2) You CANNOT add roots the way you just did. If you simplify the radicals, however, you get:
(v64)(v2) + (v25)(v2)
8v2 + 5v2
You ARE allowed to add coefficients of the radicals. Therefore, you get 13v2.

Answer 6

1. The sqrt(7/12) can be rewritten as sqrt(84)/12, or sqrt(21)/6, which is about as simple as you can get.
2. Major error here -- the sum of sqrt(128) + sqrt(50) is NOT sqrt(178). The radicals can be simplified to give 8 sqrt(2) + 5 sqrt(2), which obviously add to 13 sqrt(2).

Answer 7

=15/√2 =15*√2/√2*√2 =15√2/2 15 square root 2 over 2

Answer 8

1) 3√7/12 there is no simplified

2) √128 + √50=sqrt(2*64)+sqrt(2*25)
=8sqrt2+5sqrt2=13sqer2

Answer 9

√(a/b) = √a / √b

3 * (√7/√12)

=3*[√7/√(3*4)]

=3/2*√(7/3)

=[3√(7/3)]/2