Rationalize the denominator:?

Question

5 sq rt 6 / sq rt 10

and

1/ 1+sq rt2
How do I solve those problems?

Answer 1

5 sqrt(6)/ sqrt(10)
= 5 sqrt(3)/sqrt(5) deviding both by sqrt(2) 2 is gcd of 6 and 10

5/sqrt(5) = sqrt(5)^2/sqrt(5) = sqrt(5)

so we have = sqrt(5)*sqrt(3) = sqrt(15)

for the 2nd one

1+sqrt(2) is in denmominator so multiply both numerator and denominator by

sqrt(2) -1

numerator = sqrt(2) -1
denomenator = (sqrt(2) + 1)(sqrt(2)-1) = 2-1 =1

so value = sqrt(2) - 1

Answer 2

For #1 mult Num and Denom by sq rt 10 which gives 5 rt 60 over 10 which reduces to rt 60 over 2. The rt 60 changes to rt 4 rt 15 which becomes 2 rt 15. This reduces to rt 15 (over 1).

For #2 mult Num and Denom by 1-rt 2. This gives 1- rt 2 over -1.(1 squared minus rt2 squared) . Mult Num and Denom by -1 to get rt 2-1 ( over 1)

Answer 3

you need to make the number on the bottom a real number without a sq rt in it.

the easiest way to do this is to work out what you need to multiply the denominator by to make it a real number, in this case i would suggest root 10 since root 10xroot 10 = 10, but what you do to the bottom you must do to the top so you initally have:

5 root 6 root 10 / 10

you should be able to work out the other from that...

Answer 4

So you want to rationalize
5sqrt(6)/sqrt(10)

In this case, all you have to do is simply multiply the top and bottom by sqrt(10); the purpose of doing this is that the square root disappears when multiplied by itself, i.e. sqrt(10) * sqrt(10) = 10. Therefore, after multiplying top and bottom appropriately, we have:

[5sqrt(6)sqrt(10)]/[sqrt(10)sqrt(10)]

Which becomes

[5sqrt(6)sqrt(10)]/10

The 5 and the 10 have a common factor of 5, so we can reduce this to sqrt(6)sqrt(10)/10

Also, we can multiply square roots together, so long as we multiply what's inside them into a single square root, to get

sqrt(60)/10

-end of question 1-

To solve for question 2, i.e. to rational the denominator of

1/(1 + sqrt(2)), what you have to do is multiply by the conjugate. The conjugate is defined to be that same bracketed stuff, except with a negative sign. The reason why you want to do that is to FORM a difference of squares; after all, you do NOT want a radical on the bottom, and this is the best way to eliminate that.

In this case, we multiply the top and bottom by (1 - sqrt(2)) to get

(1 - sqrt(2)) / [(1 + sqrt(2))(1 - sqrt(2))]

Remember that (a-b)(a+b) is equal to (a^2 - b^2), a difference of squares, and that's exactly what happens above, except that the square of the square root of 2 is just 2.

(1 - sqrt(2)) / [1 - 2]
1 - sqrt(2)/(-1)
(-1) (1 - sqrt(2))
sqrt(2) - 1

Answer 5

answers:
sq rt 15
sq rt 2 - 1