SOLUTION: How do you rationalize and determine the denominator for this problem?
1 divided by 7times the square root of 5 minus 3times the square root of 3
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Question 178082: How do you rationalize and determine the denominator for this problem?
1 divided by 7times the square root of 5 minus 3times the square root of 3
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
1 divided by 7 times the square root of 5 minus 3 times the square root of 3
Depends.
What you wrote can be interpreted:
or
Second way first, because that one is easier. You need to find a multiplier that will change 
into a rational number. 
will do this quite tidily.
The process is to multiply the fractional part of your expression by 1 in the form of 
, thus:
( \frac {sqrt{5}}{sqrt{5}} ) - 3 sqrt{3})
yielding:
then apply the LCD of 35 to get:
The other interpretation is a bit trickier, but not too.
You can't simply square the denominator because you would still end up with a two term expression one term of which would still be irrational. The trick is to remember the factorization of the difference of two squares:
(a - b))
 \text { and } \math (a - b))
are called a conjugate pair. The thing is, if you multiply a binomial by its conjugate, you get the difference of two squares.
The denominator of your expression is a binomial 
whose conjugate is 
The process is to multiply the entire original fraction by 1 in the form of the conjugate of the denominator over itself, thus:
\left(\frac {7 sqrt{5} + 3 sqrt{3}}{7 sqrt{5} + 3 sqrt{3}}\right)=\left(\frac{7 sqrt{5} + 3 sqrt{3}}{245 - 27}\right)=\left(\frac{7 sqrt{5} + 3 sqrt{3}}{218}\right))
Yes, the answer is uglier than a mud fence, but because it has a rational number for a denominator, it is simpler than the original by definition.
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