2 questions.. Please help explain..?

Question

1. Is this expression false? Explain why or why not.
(square root of 2) + (square root of 6) = (square root of 8)

2. Find the area of the rectangle.
Top: square root of 3 + square root of 5
Right Side: square root of 3 + square root of 5
**This is not a typo. I'm supposed to find the area of the rectangle but it seems to me like it's a square. Please help explain.

Answer 1

1) False.
Use your calculator to come up with each value.
Here's a simpler example.
Using the same rule that sqrt(a) + sqrt(b) = sqrt(a+b)
Let a = 1 and b = 4
sqrt(1) + sqrt(4) = sqrt(1+4)
1 +2 = sqrt(5)
3 = sqrt(5)
You can see that this is NOT true.

2) The area of a rectangle is just length times width
(sqrt(3) +sqrt(5)) (sqrt(3) +sqrt(5))
Use FOIL
3 + sqrt(15) + sqrt(15) + 5
= 8 + 2sqrt(15)

It is a square, but remember that ALL squares are rectangles. They're just a special case where length = width

Answer 2

1) If we square the whole equation, we get,
2 + 6 + 2(square root of2xsquare root of 6) =8 that is,
8 + something = 8 Which can not be true.
Hence the expression is FALSE.
Is it clear?
2) THIS IS A SQUARE. The person who gave you this problem is " nuts". The area is as above.
3+5+2(square root of 3xsquare root of 5)
Is it understood?

Answer 3

NUMBER 1:
using a calculator:

square root of 2 = 1.41
square root of 6 = 2.45
square root of 8 = 2.83

which shows that (square root of 2) + (square root of 6) is NOT EQUAL to (square root of 8). Why?

because
(square root of 6) = (square root of 2) * (square root of 3) ===> since 2 * 3 is 6

while
(square root of 8) = (square root of 4) * (square root of 2), but
(square root of 4) = 2 ; thus, we can write
(square root of 8) = 2 * (square root of 2)

So, your equation:
(square root of 2) + (square root of 6) = (square root of 8)

can be simplified:
(square root of 2) + (square root of 2) * (square root of 3) = 2 * (square root of 2)

we can factor out (square root of 2) on the left side of the equation. Thus, we will have:

(square root of 2) * [ 1 + (square root of 3) ] = 2 * (square root of 2)

Both sides of the equation both have (square root of 2) in it but obviously, 2 is not equal to 1 + (square root of 3) . Therefore, your expression is FALSE.

NUMBER 2:
In geometry, all squares are rectangles but NOT all rectangle is a square. So, if the dimensions seemed to be a square, it's still a rectangle. So, the area is:

(square root of 3 + square root of 5) * (square root of 3 + square root of 5) = 3 + 2 * (square root of 3) (square root of 5) + 5, OR

AREA: 8 + 2 * (square root of 15) square units

Answer 4

1.
sqrt(2) + sqrt(6)
= sqrt(2) + sqrt(3)sqrt(2)
= sqrt(2)(1 + sqrt(3)).

sqrt(8)
= sqrt(4)sqrt(2)
= 2 sqrt(2).

If these two were equal, then you would have:
1 + sqrt(3) = 2
sqrt(3) = 2 - 1 = 1
Squaring, this gives:
3 = 1.
This is clearly false.

The expressions are therefore not equal.

2.
The rectangle is a square, and its area is:
(sqrt(3) + sqrt(5))^2
= 3 + 5 + 2sqrt(15)
= 8 + 2sqrt(15)
= 2(4 + sqrt(15)).

Answer 5

. Yes, it is false.
sqrt(2)+sqrt(6) = sqrt(8)
sqrt(2) +sqrt(2)*sqrt(3) = 2sqrt(2)
sqrt(2)[1+sqrt(3)] = 2sqrt(2)
1+sqrt(3) = sqrt(2) which is obviously false.

Yes, it is a square (which is also a rectangle)
A = [sqrt(3) +sqrt(5)]^2= 3 + 5 + 2sqrt(15)= 8+2sqrt(15)

Answer 6

sqrt(2) + sqrt(6) = 2sqrt(2), so sqrt(6) > sqrt(2), so eqn was false.

square is a rectangle with equal sides, so ok.
[sqrt(3)+ sqrt(5)]^2 = 3 + 5 + 2sqrt(3)sqrt(5) = 8 + 2sqrt(15)