2/1+√2-2/2+√2
Without using calculator or mathematical table
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.
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.