Question 86811
Since 500 is an even number you know that it is divisible by 2.  Therefore, you can write
it as {{{500 = 250 * 2}}}
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But 250 is also even and divisible by 2. So 250 can be replaced by 125 * 2. Therefore,
you can replace 500 by 125*2*2 which is equivalent to {{{500 = 125*2^2}}}
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Then {{{125 = 5*5*5 = 5*5^2}}}. So if you substitute {{{5*5^2}}} for 125 you get:
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{{{500 = 5*5^2*2^2}}}
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Taking the square root of all three terms in this multiplication string results in:
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{{{sqrt(500)= sqrt(5*5^2*2^2) = sqrt(5)*sqrt(5^2)*sqrt(2^2) = sqrt(5)*5*2 = 10*sqrt(5)}}}
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So the simplified form of {{{sqrt(500)}}} is  {{{10*sqrt(5)}}}
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Had you recognized it earlier, you could have written:
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{{{sqrt(500) = sqrt(100*5) = sqrt(100)*sqrt(5) = 10*sqrt(5)}}}
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much quicker, but not as much practice as the long way.
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Hope this helps you to see how square root problems such as this one can be simplified.
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