Simplifying the Square Root of a Sum
Date: 09/27/2004 at 18:36:20 From: Alyssa Subject: Why does sqrt(a + b) not equal the sqrt(a) + sqrt(b) Why is it that sqrt(a + b) does not equal sqrt(a) + sqrt(b)? I.e., the sqrt(36 + 49) = sqrt(85) or about 9.2195. But the sqrt(36) + sqrt(49) = 13. This confuses me because I know that the sqrt(ab) = sqrt(a)sqrt(b). I don't understand why under the operations of division or multiplication the sqrt can be separated out but in the operations of addition or subtraction it can not.
Date: 09/27/2004 at 19:01:08 From: Doctor Schwa Subject: Re: Why does sqrt(a+b) not equal the sqrt(a) + sqrt(b) Hi Alyssa, Maybe it would help to think of things this way: a * (b + c) = a * b + a * c for all numbers, because multiplication is repeated addition. If a = 3, you could write out 3*(b+c) = b+c + b+c + b+c and then simplify to get 3b + 3c because it's all addition. Similarly, (a * b)^c = a^c * b^c because exponents are repeated multiplication. If c = 3, you could write out (a*b)^3 = a*b * a*b * a*b and then simplify to get a^3 * b^3 because it's all multiplication. But (a + b)^c is NOT the same as a^c + b^c because exponents aren't repeated addition. If c = 3, you could write out (a+b)^3 = (a+b) * (a+b) * (a+b) but then you can't simplify the * and the + mixture! In much the same way, sqrt can be thought of as an exponent (it's the 1/2 power), so sqrt(a*b) = sqrt(a) * sqrt(b) but sqrt(a+b) can't be rearranged in any nice way. Does that help it make more sense? - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/
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