First of all, let's look at the numbers. sqrt(36 + 64) = sqrt(100) = 10.
sqrt(36) + sqrt(64) = 6 + 8 = 14.
So you see those aren't the same number. That's why you can't just replace one expression by the other, if that's what you're asking.
Or are you asking WHY they aren't the same number? Well, remember the Pythagorean Theorem? a^2 + b^2 = c^2? a^2 + b^2 is not equal to (a + b)^2.
That is, c^2 is not equal to (a + b)^2. The hypotenuse of a triangle c is not just the sum of the other two sides. The distance diagonally across the triangle is shorter than the distance along the two legs. Those are not the same distance.
It's related to that. Let 36 = a^2, let 64 = b^2. Then you're asking about sqrt(a^2 + b^2) which is sqrt(c^2), compared to sqrt(a^2) + sqrt(b^2). That is, you're asking why c is not equal to a + b.
They are different because in the former, you are not taking the square root of 64, as it has no sign. Therefore, it is 6+64 = 70. In the latter, you are taking the square root, so it is 6+8=14.
On the other hand, if you meant does â(36+64) not equal â36+â64, the answer is because of the order of operations. In accordance with proper mathematical procedure, anything inside brackets must be calculated first. So in the former, you are adding 36 to 64, and then taking the root of that summation. The answer will be ten. in the latter, you are taking their roots individually, and then adding, so you get 14.
Let's take the square root sign to be a bracket; and you know in BODMAS, the B which happens mean Bracket; so let's go back to ur question.............. The first quation; you will have to add 36 + 64 up, then whateva the answer is, you find its square root... For the second question; you find the square root for 36 n 64 up then u add it up... I.e
36+64 =100, then the square root of 100 = 10...... Square root of 36 is 6, while that of 64 is 8... When u sum up 6 and 8 you have 14.... Hope u now understand it...:)
So one way is to say that sqrt(36+64) = sqrt(100) = 10 but sqrt(36) + sqrt(64) = 6 + 8 which is not equivalent to 10.
The real way is to show that f(a+b) is only equal to f(a) + f(b) if f is a "linear operator". Because sqrt can be expressed as an exponential function with power of 1/2, it is not a linear function and therefore sqrt(a+b) does not equal sqrt(a) + sqrt(b)
Answers & Comments
Verified answer
√(a^2 + b^2) = √[(a + b)^2 - 2ab]
√a^2 + √b^2 = a + b
√[(a + b)^2 - 2ab] = a + b
{√[(a + b)^2 - 2ab]}^2 = (a + b)^2
(a + b)^2 - 2ab = (a + b)^2
(a + b)^2 - (a + b)^2 = 2ab
0 = 2ab => this true only if a = 0 or b = 0 or a = b = 0, otherwise false.
First of all, let's look at the numbers. sqrt(36 + 64) = sqrt(100) = 10.
sqrt(36) + sqrt(64) = 6 + 8 = 14.
So you see those aren't the same number. That's why you can't just replace one expression by the other, if that's what you're asking.
Or are you asking WHY they aren't the same number? Well, remember the Pythagorean Theorem? a^2 + b^2 = c^2? a^2 + b^2 is not equal to (a + b)^2.
That is, c^2 is not equal to (a + b)^2. The hypotenuse of a triangle c is not just the sum of the other two sides. The distance diagonally across the triangle is shorter than the distance along the two legs. Those are not the same distance.
It's related to that. Let 36 = a^2, let 64 = b^2. Then you're asking about sqrt(a^2 + b^2) which is sqrt(c^2), compared to sqrt(a^2) + sqrt(b^2). That is, you're asking why c is not equal to a + b.
Does that make sense?
They are different because in the former, you are not taking the square root of 64, as it has no sign. Therefore, it is 6+64 = 70. In the latter, you are taking the square root, so it is 6+8=14.
On the other hand, if you meant does â(36+64) not equal â36+â64, the answer is because of the order of operations. In accordance with proper mathematical procedure, anything inside brackets must be calculated first. So in the former, you are adding 36 to 64, and then taking the root of that summation. The answer will be ten. in the latter, you are taking their roots individually, and then adding, so you get 14.
Let's take the square root sign to be a bracket; and you know in BODMAS, the B which happens mean Bracket; so let's go back to ur question.............. The first quation; you will have to add 36 + 64 up, then whateva the answer is, you find its square root... For the second question; you find the square root for 36 n 64 up then u add it up... I.e
36+64 =100, then the square root of 100 = 10...... Square root of 36 is 6, while that of 64 is 8... When u sum up 6 and 8 you have 14.... Hope u now understand it...:)
So one way is to say that sqrt(36+64) = sqrt(100) = 10 but sqrt(36) + sqrt(64) = 6 + 8 which is not equivalent to 10.
The real way is to show that f(a+b) is only equal to f(a) + f(b) if f is a "linear operator". Because sqrt can be expressed as an exponential function with power of 1/2, it is not a linear function and therefore sqrt(a+b) does not equal sqrt(a) + sqrt(b)