Question

Why this Equation
$\sqrt{7 } - \sqrt{5} = \sqrt{2 }$
is Wrong? Explain.

Answer 1

Because in an equation there should be constant and variable this equation should be like this
underrout 7x- underrout 5y=underrout 2

Answer 2

simplw statement: This is wrong because
$\sqrt{7} = 2.645 \\ \sqrt{5} = 2.236$
And when these two guys are subtracted from each other the result isn't equal to 1.414 which is value of root 2.

Now let's come to the explanation why this happened,

When we subtract one number from another say for example
5-3

Let's say 5 = | | | | | (5 lines)
and 3 = | | |

what we basically do here (in subtraction) is take away, those three lines from those 5 line,

| | | | | - | | |
and so you'll end up with | | which means 2 lines i.e representing 2

Now,

A number is under root that means how will you break it in order to get two identical (talking airthematically equal) numbers.

So if you have
√100

then you gotta break it in two equal numbers in order to get it out of root.

and
the result must be
√(10 ×10) (which equals 10)

which could also be written as
√10 × √ 10

square root means that number raised to the power of 1/2

so, our number becomes,

${10}^{\frac{1}{2} } \times {10}^{ \frac{1}{2} }$

Using expenential properties,
when bases are same and numbers are in multiplication with each other then you just simply add those numbers in power and here we get

${10}^{( \frac{1}{2} + \frac{1}{2} )} \\ = {10}^{ \frac{2}{2} } \\ = {10}^{1} \\ = 10$

Now what happens in case of √5 is you cannot break it down into two equal numbers as in the case of √100
but we still try to do that and end up with approximate values.

As I've written above that √5 = 2.236
which means when you multiple 2.236 with itself you'll end up with a number approximately equal to 5 and this accuracy increases as you increases the numbers after decimal behind.

Now back to your question,


So we have

√7 - √5

so, now (I hope) you can understand that √7 isn't exactly something like 7 nor is √5 and so their difference isn't going to be equal to √2.

subtracting
√5 from √7 here means subtracting those similar 2.236 lines from 2.645 lines, which definately isn't equal to 1.414 lines.




Hope you got it.