how do I simplify radicals with fractional radicands?

ex. 7sqrt(2) + sqrt(50) - 2sqrt(18)

10sqrt(3/5) - 24sqrt(5/3)

3sqrt(2/9) + 1/2sqrt(32) + sqrt(9/8)


This is an exercise in recognizing numbers. When I look at the first problem what I see is that

50 = 25 2 and 18 = 9 2. That is, each of them is a square times 2. Thus sqrt(50) = sqrt(25 2) = 5 sqrt(2) and sqrt(18) = sqrt(9 2) = 3 sqrt(2)

In the second problem I would like the denominators to be squares so that I can rewrite the expression with no fractions under the square root symbol. I can do this if I write

For the third problem the key is 32 since 32 = 16  2, again a square times 2. Can you write each of the other two radicands as a square times 2?


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