SOLUTION: Does the square root of a plus the square root of b equal the sum of the square root of a+b? Why or why not? Thank you for any help in answering this question. Daniel H. Jack

Click here to see ALL problems on Miscellaneous Word Problems

Question 1015476: Does the square root of a plus the square root of b equal the sum of the square root of a+b? Why or why not?
Thank you for any help in answering this question.
Daniel H. Jack
Found 2 solutions by Alan3354, Theo:
Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website!
Does the square root of a plus the square root of b equal the sum of the square root of a+b? Why or why not?
----------
?
1 counter example disproves this.
?
?
?
They're not equal.

Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
no.

the square root of a * the square root of b = the square root of (a*b), but.....

the square root of a + the square root of b does not equal the square root of (a + b).

consider square root of 16 plus the square root of 25.

square root of 16 is 4.
square root of 25 is 5.
4 + 5 = 9

now consider square root of (16 + 25)
that becomes square root of (41) which is equal to 6.403124237 which is not equal to 9.

however,

square root of 16 * square root of 25 is equal to 4 * 5 = 20
square root of (16 * 25) = square root of (400) = 20

it works with multiplication.
it doesn't work with addition.

all you need is one example where it doesn't works to disprove the statement.

with sqrt(a) * sqrt(b) = sqrt(a*b), you won't find one no matter how many times you try.

with sqrt(a) + sqrt(b) = sqrt(a+b), you wan find one very easily as above.

the formal proof is probably a lot more complicated so i'll leave that one alone.

the best you can do with sqrt(a) + sqrt(b) is factor out the common terms.

for example:

sqrt(2 * 4) + sqrt(2 * 5) = sqrt(2) * sqrt(4) + sqrt(2) * sqrt(5).

you can factor out the common term of sqrt(2) to get:

sqrt(2 * 4) + sqrt(2 * 5) = sqrt(2) * (sqrt(4) + sqrt(5)

since you know that sqrt(4) = 2, then you can simplify further to get:

sqrt(2 * 4) + sqrt(2 * 5) = sqrt(2) * (2 + sqrt(5)

you can confirm your simplification is good by comparing the result of the original equation with the result of the final equation.

sqrt(2 * 4) + sqrt(2 * 5) = sqrt(8) + sqrt(10) = 5.990704785

sqrt(2) * (2 + sqrt(5)) = 5.990704785

they're the same so your simplification is good.

bottom line:

sqrt(a) * sqrt(b) equals sqrt(a*b)

sqrt(a) + sqrt(b) does not equal sqrt(a + b)

here's a reference that should help you to understand.

http://www.purplemath.com/modules/radicals.htm