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Complicated formula

Problem Solving — algebra and arithmetic
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Senior | Next Rank: 100 Posts
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by r_walid » Wed Nov 04, 2009 10:14 am
what is the following equal to?

sqrt [ 2xsqrt(63) + 2/(8 + 3xsqrt(7) ) ]


the answer isn't as important as to find a way to quickly solve this. I know there is a trick, but i can't seem to find it.

thanks
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by mp2437 » Wed Nov 04, 2009 10:56 am
Answer is 4. I don't see a shorter way to solve this then by figuring out like terms, and then rationalizing the denominator.

Step 1: Look at what is on the inside. 2*sqrt(63) can be rewritten as 6*sqrt(7), and when you add that to 2/(8+3*sqrt(7)), you need to have a common denominator or 8+3*sqrt(7).

Step 2: from step 1, you will get the following term:

[(6*sqrt(7)*(8+3*sqrt(7)) + 2]/(8+3*sqrt(7))

This comes out to (48*sqrt(7) + 128)/(8+3*sqrt(7)).

Step 3: Now you need to rationalize denominator, so multiply both the numerator and denominator by (8-3*sqrt(7)).

This gives you (48*sqrt(7) + 128)*(8 - 3*sqrt(7)) / 1. You get 1 in the denominator after the rationalization (8+3*sqrt(7))*(8-3*sqrt(7)) = 64 - 3*3*7 = 64 - 63 = 1

Step 4: Doing in the arithmetic in the numerator will give you a numerator of 16 (go carefully). So final step is that you have square root over the entire term, so sqrt(16) = 4.
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by r_walid » Wed Nov 04, 2009 11:08 am
mp2437 wrote:Answer is 4. I don't see a shorter way to solve this then by figuring out like terms, and then rationalizing the denominator.

Step 1: Look at what is on the inside. 2*sqrt(63) can be rewritten as 6*sqrt(7), and when you add that to 2/(8+3*sqrt(7)), you need to have a common denominator or 8+3*sqrt(7).

Step 2: from step 1, you will get the following term:

[(6*sqrt(7)*(8+3*sqrt(7)) + 2]/(8+3*sqrt(7))

This comes out to (48*sqrt(7) + 128)/(8+3*sqrt(7)).

Step 3: Now you need to rationalize denominator, so multiply both the numerator and denominator by (8-3*sqrt(7)).

This gives you (48*sqrt(7) + 128)*(8 - 3*sqrt(7)) / 1. You get 1 in the denominator after the rationalization (8+3*sqrt(7))*(8-3*sqrt(7)) = 64 - 3*3*7 = 64 - 63 = 1

Step 4: Doing in the arithmetic in the numerator will give you a numerator of 16 (go carefully). So final step is that you have square root over the entire term, so sqrt(16) = 4.


Thank you. but don't you think there has to be a faster way? step 4 takes forever to do, there has to be a trick that will help solve step 4 in a fast way. any thoughts anyone?
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