Evaluate 3/2√2+3-√2/2√2-3
To evaluate the expression, we can simplify the numerator and the denominator separately, and then perform the division.
1) Simplifying the numerator:
We have 3/2√2 + 3 - √2/2√2
The first term, 3/2√2, is already simplified.
The second term, 3, is just a constant and does not need any simplification.
The third term, √2/2√2, can be simplified by multiplying the numerator and denominator by √2 to get rid of the radical in the denominator. This gives us (√2)(√2)/(2√2)(√2) = 2/4 = 1/2.
So our numerator becomes: 3/2√2 + 3 - 1/2.
2) Simplifying the denominator:
The denominator, 2√2 - 3, does not require any further simplification.
3) Putting it all together:
Our expression now becomes: (3/2√2 + 3 - 1/2)/(2√2 - 3)
Now, we can multiply the numerator and denominator by the conjugate of the denominator, which is 2√2 + 3, to eliminate any radicals in the denominator.
(3/2√2 + 3 - 1/2)(2√2 + 3)/(2√2 - 3)(2√2 + 3)
Expanding the numerator and the denominator, we get:
(3(2√2 + 3) + 3(2√2 + 3) - 1(2√2 + 3))/(4(2)-9)
Simplifying further:
(6√2 + 9 + 6√2 + 9 - 2√2 - 3)/(8-9)
Combining like terms:
(10√2 + 15)/(8-9)
(10√2 + 15)/(-1)
-10√2 -15
So the final value of the expression 3/2√2 + 3 - √2/2√2 - 3 is -10√2 - 15.
To evaluate the expression 3/2√2 + 3 - √2/2√2 - 3, let's break it down step-by-step:
Step 1: Simplify the terms in the numerator and denominator of each fraction separately.
3/2√2: No simplification is needed.
√2/2√2: The numerator and denominator both have a common factor of √2. Dividing both by √2 gives us 1/2.
The expression now becomes: 3/2√2 + 3 - 1/2 - 3
Step 2: Combine the like terms by finding a common denominator for the fractions.
The common denominator is 2√2, since both 2√2 and 2 have this as a factor.
The expression now becomes: (3+3*2√2)/2√2 - 1/2 - 3
Step 3: Simplify the numerator.
3+3*2√2 simplifies to 3+6√2, or 6√2+3.
The expression now becomes: (6√2+3)/2√2 - 1/2 - 3
Step 4: Combine the terms using the common denominator.
The common denominator is still 2√2.
The expression now becomes: (6√2+3)/2√2 - (1*√2)/2√2 - (3*2√2)/2√2
Step 5: Simplify each fraction by dividing numerator and denominator by their common factor.
(6√2+3)/2√2: No simplification is needed.
(1*√2)/2√2: The numerator and denominator both have a common factor of √2. Dividing both by √2 gives us 1/2.
(3*2√2)/2√2: The numerator and denominator both have a common factor of 2√2. Dividing both by 2√2 gives us 3/2.
The expression now becomes: (6√2+3)/2√2 - 1/2 - 3/2
Step 6: Simplify further by finding a common denominator for the fractions.
The common denominator is 2.
The expression now becomes: (6√2+3)/2√2 - (1*√2)/2 - (3*1)/2
Step 7: Simplify each fraction by dividing numerator and denominator by their common factor.
(6√2+3)/2√2: No simplification is needed.
(1*√2)/2: The numerator and denominator both have a common factor of 2. Dividing both by 2 gives us √2/1, or simply √2.
(3*1)/2: The numerator and denominator both have a common factor of 2. Dividing both by 2 gives us 3/2.
The expression now becomes: (6√2+3)/2√2 - √2/2 - 3/2
Step 8: Combine the terms using the common denominator.
The common denominator is 2√2.
The expression now becomes: [(6√2+3) - (√2) - (3)]/2√2
Step 9: Simplify the numerator.
(6√2+3) - (√2) - (3) simplifies to 6√2.
The final expression is: 6√2/2√2
Step 10: Simplify the fraction.
6√2/2√2 = 6/2 = 3
Therefore, the expression evaluates to 3.