2/1+√2-2/2+√2
Without using calculator or mathematical table
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.
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
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)
To simplify the expression 2/1 + √2 - 2/2 + √2 without using a calculator or mathematical table, follow these steps:
Step 1: Simplify the fractions in the expression
- 2/1 is the same as 2
- 2/2 is equal to 1
So, the expression becomes:
2 + √2 - 1 + √2
Step 2: Combine like terms
Combine the numbers and the square root terms separately:
2 + 1 - 1 + √2 + √2
Step 3: Simplify the numbers
2 + 1 - 1 = 2
Step 4: Simplify the square root terms
√2 + √2 = 2√2
Step 5: Combine the simplified terms
2 + 2√2
So, the simplified form of the expression 2/1 + √2 - 2/2 + √2 is 2 + 2√2.