3√2-1/√2-1/√8
To simplify the given expression, we can start by rationalizing the denominators of the fractions.
Recall that the conjugate of a denominator is formed by changing the sign between the terms.
For the first fraction, we have:
3√2 - 1/√2
To rationalize the denominator, multiply both the numerator and denominator by the conjugate of √2, which is (-√2):
[3√2 * (-√2) - 1 * (-√2)] / (√2 * (-√2))
Simplifying this expression, we get:
(-6 - √2) / (-2)
Now, let's focus on the second fraction:
1/√8
To rationalize the denominator, multiply both the numerator and denominator by the conjugate of √8, which is (√8):
[1 * (√8)] / (√8 * (√8))
Simplifying this expression, we get:
√8 / 8
Now that we have rationalized the denominators, we can combine the fractions:
(-6 - √2)/(-2) - (√8)/8
To subtract these fractions, we need a common denominator. The common denominator in this case is 8:
[(-6 - √2) * 4 - √8] / 8
Expanding the numerator, we get:
(-24 - 4√2 - √8) / 8
Next, we can simplify the terms inside the square roots:
-24 - 4√2 - 2√2
Combining like terms, we get:
-24 - 6√2
Therefore, the simplified form of 3√2 - 1/√2 - 1/√8 is -24 - 6√2.
To simplify the expression 3√2 - 1/√2 - 1/√8, we need to rationalize the denominators.
Step 1: Simplify the expression within the radicals:
3√2 = 3√(2/1) = 3√2/√1 = 3√2
Step 2: Rationalize the denominator of the second term:
1/√2 = (1/√2) * (√2/√2) = √2/2
Step 3: Rationalize the denominator of the third term:
1/√8 = (1/√8) * (√8/√8) = √8/8 = √2/2
Now, the expression becomes:
3√2 - √2/2 - √2/2
Step 4: Combine like terms:
= 3√2 - (√2/2 + √2/2)
= 3√2 - 2√2/2
= (6√2 - 2√2)/2
= 4√2/2
= 2√2
To simplify the expression 3√2 - 1/√2 - 1/√8, we can follow these steps:
Step 1: Simplify the denominators:
- The square root of 2 cannot be simplified further.
- The square root of 8 can be simplified using the property √(a * b) = √a * √b.
So, √8 = √(4 * 2) = √4 * √2 = 2 * √2 = 2√2.
Substituting these simplified values into the original expression, we have:
3√2 - 1/√2 - 1/2√2
Step 2: Combine like terms:
- The terms with √2 in both the numerator and denominator can be combined.
To add or subtract such terms, their denominators must be the same.
Since the denominator of 1/√2 is √2, we need to express the other term, 2√2, with the same denominator.
We can do this by multiplying the numerator and denominator by √2:
3√2 - 1/√2 - 1/2√2 = 3√2 - (1 * √2)/(√2 * √2) - 1/(2 * √2)
Simplifying further:
= 3√2 - √2/2 - 1/(2√2)
Step 3: Find a common denominator:
- To add or subtract fractions, we need a common denominator.
The common denominator for 2 and 2√2 is 2 * √2.
Multiplying the numerator and denominator of the second fraction (√2/2) by √2 will give us the common denominator:
3√2 - (1 * √2)/(√2 * √2) - 1/(2 * √2)
= 3√2 - (√2 * √2)/(2 * √2 * √2) - 1/(2 * √2)
Simplifying further:
= 3√2 - 2/ (2√2) - 1/(2√2)
Step 4: Combine like terms:
- The terms with √2 in the denominator can be combined now that they have the same denominator:
3√2 - 2/ (2√2) - 1/(2√2)
= 3√2 - (2 + 1)/(2√2)
Simplifying the numerator:
= 3√2 - 3/(2√2)
Step 5: Combine the terms:
3√2 - 3/(2√2)
= (3√2 * 2√2 - 3) / (2√2) (applying the concept a/b - c/d = (a*d - b*c)/(b*d))
Simplifying the numerator:
= (6 - 3) / (2√2)
= 3 / (2√2)
This is the simplified form of the expression 3√2 - 1/√2 - 1/√8.