Rationalize the denominator?

Question

5 times the square root of 5 SUBTRACT 4 times the square root of 3

ALL THAT under 1 (as a fraction)

Answer 1

1/5rt5-4rt3
=5rt5+4rt3/125-48
==5rt5+4rt3/77

Answer 2

1 / (5√5 - 4√3)

When the denominator looks like a+b and there's a square root involved, multiply top and bottom by the conjugate (a-b) to get rid of the square root:

(1 / (5√5 - 4√3)) * (5√5 + 4√3)/(5√5 + 4√3)) =
(5√5 + 4√3) / ((5√5 - 4√3)(5√5 + 4√3)) =
(5√5 + 4√3) / ((5√5)^2 - (4√3)(5√5) + (4√3)(5√5) - (4√3)^2) =
(5√5 + 4√3) / (5*25 - 16*3) =
(5√5 + 4√3) / (125 - 48) =
(5√5 + 4√3) / 77

Answer 3

Multiply top and bottom by (5 x 5^½ + 4 x 3^½)
= (5 x 5^½ + 4 x 3^½)/(125 - 48)
= (5x5^½ + 4x3^½) / 77
p.s. I would like to know how geezerbill shows the square root of a number by using a square root sign as opposed to using ^1/2

Answer 4

To rationalize the denominator, you need to multiply both the numerator and the denominator by the conjugate form of the denominator. That is, since you have:

(5 * sqrt(5)) - (4 * sqrt(3))

The conjugate form is

(5 * sqrt(5)) + (4 * sqrt(3))

Then, your numerator will be:

(5 * sqrt(5)) + (4 * sqrt(3))

And the denominator will be the conjugate times what you had originally, or:

(5sqrt(5)*5sqrt(5)) + (5sqrt(5)4sqrt(3)) - (5sqrt(5)4sqrt(3)) - (4sqrt(3)4sqrt(3))

This simplifies to:
(5sqrt(5)*5sqrt(5)) - (4sqrt(3)4sqrt(3))
(5 * 5 * sqrt(5) * sqrt(5)) - (4 * 4 * sqrt(3) * sqrt(3))
(125) - (48)
77

So, your final fraction is:

(5 * sqrt(5)) + (4 * sqrt(3))
----------------------------------
77

Answer 5

1/[(5√5)-(4√3)]
=1/[(5√5)-(4√3)] * [(5√5)+(4√3)] /[(5√5)+(4√3)]
=[(5√5)+(4√3)] /[(5√5^2)-(4√3^2)
=5√5+4√3/125+48
=5√5+4√3/173

Answer 6

you multiply by [5(sqr5) + 4(sqr3)]/[5(sqr5) + 4(sqr3)] to rationalise the denominator and you'll get

[5(sqr5) + 4(sqr3)]/77

Answer 7

1/(5sqrt(5) - 4sqrt(3)) * (5sqrt(5) + 4sqrt(3))/(5sqrt(5) + 4sqrt(3))
= (5sqrt(5) + 4sqrt(3))/77

Answer 8

............1............
--------------------- ???
5sqrt(5)-4sqrt(3)

You need to multiply by 5sqrt(5)+4sqrt(3) OVER 5sqrt(5)+4sqrt(3)
You get
5sqrt(5)+4sqrt(3)
----------------------
25*5-16*3

Which is equal to

5sqrt(5)+4sqrt(3)
----------------------
...........77.............