Question 95958
Given to simplify:
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{{{3*sqrt(45)-2*sqrt(125)}}}
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On a calculator you can find that the square root of 45 is 6.708203932 so multiplying 
the square root of 45 times 3 results in 3*6.708203932 = 20.1246118. Substituting this number
into the problem in place of {{{3*sqrt(45)}}} reduces the problem to:
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{{{20.1246118 - 2*sqrt(125)}}}
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Next you can use the calculator to find that {{{sqrt(125) = 11.18033989}}}. Multiplying this
by -2 results in {{{-2*sqrt(125) = -2*11.18033989 = -22.36067977}}}. Then substituting this 
into the equation reduces the equation to:
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{{{20.1246118 - 22.36067977 = -2.236067975}}}
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So the answer to this problem is -2.236067975 within the roundoff error on the last decimal 
place.
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Now let's work the problem without doing it on a calculator. Start with:
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{{{3*sqrt(45)-2*sqrt(125)}}}
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Note that by the rules of radicals, {{{sqrt(45) = sqrt(9*5) = sqrt(9)*sqrt(5) = 3*sqrt(5)}}}
So you can substitute {{{3*sqrt(5)}}} for {{{sqrt(45)}}} to get:
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{{{3*3*sqrt(5) - 2*sqrt(125)}}}
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Now let's simplify {{{sqrt(125)}}}:
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{{{sqrt(125) = sqrt(25*5) = sqrt(25)*sqrt(5)= 5*sqrt(5)}}}
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You can now substitute {{{5*sqrt(5)}}} for {{{sqrt(125)}}} to get:
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{{{3*3*sqrt(5) - 2*5*sqrt(5)}}}
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Do the multiplications. 3*3 = 9 and -2*5 = -10. So this becomes:
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{{{9*sqrt(5) - 10*sqrt(5)}}}
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You can factor {{{sqrt(5)}}} from each term to get:
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{{{(9-10)*sqrt(5)}}}
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Combine the two numbers in the parentheses to get:
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{{{(-1)*sqrt(5) = -1*sqrt(5) = -sqrt(5)}}}
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So the answer we got is {{{-sqrt(5)}}}. If you find the square root of 5 on a calculator
you will find that it is 2.236067977. Therefore, our answer becomes:
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{{{-sqrt(5) = -2.236067977}}}
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This agrees with the answer we got by doing all the work on a calculator. So you can use
either method depending on what is compatible with the way your teacher wants it done.
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Hope this helps you to understand the problem and how to get the answer.
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