SOLUTION: sqrt3 ______=? it says to rationalize each denominator thanks for your help! sqrt80x

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Question 637290: sqrt3
______=? it says to rationalize each denominator thanks for your help!
sqrt80x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
First of all, please post your questions in an appropriate category. Many tutors will only for for problems in the categories they like. So those who like square root problems would have never seen this problem since you posted it under "exponential and logarithmic equations". It is to your advantage to post in an appropriate category.

Second, please don't try to use fraction bars on algebra.com. They do not display well and it is often very difficult to understand them. Use "/" instead:
sqrt(3)/sqrt(80x)

sqrt%283%29%2Fsqrt%2880x%29
There are several ways to rationalize a denominator. For a problem like this I prefer to...
1. Use the root%28a%2C+p%29%2Froot%28a%2C+q%29+=+root%28a%2C+p%2Fq%29 property of radicals to merge the two square roots into one:
sqrt%283%2F80x%29

2. Make the denominator a perfect square. This can be done by reducing the fraction and/or multiplying the numerator and denominator by the same expression. Our fraction does not reduce. So we will have to multiply. But by what? Well, we want a perfect square it would seem that 80x is what we should multiply by. 80x can be used and it will give us the correct answer.

But there is a simpler solution. All we need is any prefect square. We do not need %2880x%29%5E2. If you look at 80x as 16*5*x then you can see that another factor of 5 and another factor of x will turn 80x into a perfect square. So we are going to multiply the numerator and denominator by 5x instead of 80x:
sqrt%28%283%2F80x%29%285x%2F5x%29%29
which simplifies to:
sqrt%2815x%2F400x%5E2%29

3. Use the root%28a%2C+p%29%2Froot%28a%2C+q%29+=+root%28a%2C+p%2Fq%29 property of radicals again, this time in the opposite direction, to split the square root into two:
sqrt%2815x%29%2Fsqrt%28400x%5E2%29

4. Simplify the two square roots. If you've done step 2 correctly then the square root in the denominator should disappear. The square root in the numerator may or may not simplify. Ours does not:
sqrt%2815x%29%2F20x
This is the simplified expression with a rational denominator.

P.S. In case you are curious, here is how the problem would have worked if we did not figure out that 5x worked better than 80x to get a perfect square at step 2.
sqrt%283%29%2Fsqrt%2880x%29
sqrt%283%2F80x%29
sqrt%28%283%2F80x%29%2880x%2F80x%29%29
sqrt%28240x%2F6400x%5E2%29
sqrt%28240x%29%2Fsqrt%286400x%5E2%29
Using the 80x instead of a 5x we get a square root in the numerator that will simplify...
sqrt%2816%2A15x%29%2F80x
sqrt%2816%29%2Asqrt%2815x%29%2F80x
4%2Asqrt%2815x%29%2F80x
... and we get a fraction that will reduce:
4%2Asqrt%2815x%29%2F%284%2A20x%29
cross%284%29%2Asqrt%2815x%29%2F%28cross%284%29%2A20x%29
sqrt%2815x%29%2F20x