2/1+√2-2/2+√2

Question

Without using calculator or mathematical table

Answer 1

To simplify the expression, we will start by simplifying the denominators.

In the first fraction, the denominator is 1.

In the second fraction, the denominator is 2+√2.

Since these denominators are different, we need to find a common denominator.

To find a common denominator, we can multiply the first denominator by (2+√2) and the second denominator by 1.

This gives us:

2/((2+√2)(1)) + (√2 - 2)/((2+√2)(2+√2))

Simplifying this further, we get:

2/(2+√2) + (√2 - 2)/(4+2√2+2√2+2)

2/(2+√2) + (√2 - 2)/(6+4√2)

Now, we can add the fractions together by finding a common denominator.

The common denominator is (2+√2)(6+4√2).

So, we have:

(2(6+4√2))/(2+√2)(6+4√2) + ((√2 - 2)(2+√2))/(2+√2)(6+4√2)

Simplifying:

(12+8√2)/(12+8√2+2√2+√2) + (2√2 + √2^2 - 4√2 - 2)/(12+8√2+2√2+√2)

(12+8√2)/(12+10√2) + (2√2 + 2 - 4√2 - 2)/(12+10√2)

Now, we can combine the like terms:

(12+8√2)/(12+10√2) + (-2√2)/(12+10√2)

Now, we can add the numerators together:

(12+8√2 - 2√2)/(12+10√2)

Simplifying:

(12+6√2)/(12+10√2)

That is the simplified form of the expression.

Answer 2

I get so tired of people who are careless with parentheses.
As written, a denominator of 1 is useless, so I assume they meant

2/(1+√2) - 2/(2+√2)
= (2(2+√2) - 2(1+√2)) / ((1+√2)(2+√2))
= 2 / (4+3√2)
= 3√2 - 4

Answer 3

Apologies for the confusion caused by the original response. You are correct, the correct interpretation of the expression is:

2/(1+√2) - 2/(2+√2)

To simplify, we can find a common denominator:

((2(2+√2)) - (2(1+√2))) / ((1+√2)(2+√2))

Simplifying further:

(4+2√2 - 2 - 2√2)) / ((1+√2)(2+√2))

Combining like terms:

(2√2 + 2) / ((1+√2)(2+√2))

Now, we can multiply the conjugate of the denominator to simplify:

(2√2 + 2) / (2+√2)(2+√2))

Expanding the denominator:

(2√2 + 2) / (4 + 2√2 + 2√2 + √2*√2)

Simplifying further:

(2√2 + 2) / (4 + 4√2 + 2)

Combining like terms:

(2√2 + 2) / (6 + 4√2)

The expression is simplified as:

(2√2 + 2) / (6 + 4√2)