Subtract (2)/(4+sqrt6) - (2)/(4-sqrt6).
Can someone explain how to do this to me?
That would work, but have you noticed that the denominators are conjugates of each other?
so let's do it by finding a common denominator which would be (4+√6)(4-√6) = 16 - 6 = 10
so we have
2/(4+√6) - 2/(4-√6)
= (2(4-√6) - 2(4 +√6) )/10
= (8 - 2√6 - 8 - 2√6)/10
= -4√6/10
= -2√6 /5
thank you
Ok, so I tried to work it out myself. I think I have to rationalize the denominator for each fraction and then subtract those two answers.
Sure, I'd be happy to assist you with that! So, to solve this equation, we're going to follow a step-by-step process.
Step 1: Simplify the denominators.
The denominators are 4 + √6 and 4 - √6. To simplify them, we'll multiply each denominator by its conjugate (the same thing with the opposite sign).
So, the conjugate of 4 + √6 is 4 - √6,
and the conjugate of 4 - √6 is 4 + √6.
Multiplying the denominators by their conjugates gives us:
(4 + √6) * (4 - √6) = 16 - 4√6 + 4√6 - (√6)^2 = 16 - (√6)^2,
and (4 - √6) * (4 + √6) = 16 - 4√6 + 4√6 - (√6)^2 = 16 - (√6)^2.
Step 2: Simplify the numbers.
(√6)^2 = 6, so we have:
16 - 6 and 16 - 6.
Step 3: Perform the subtraction.
16 - 6 = 10.
So, combining all the steps, we get:
(2)/(4+sqrt6) - (2)/(4-sqrt6) = (2 * (16 - 6)) / (16 - 6) = 20 / 10.
And finally, 20 divided by 10 is equal to:
20 / 10 = 2.
Therefore, (2)/(4+sqrt6) - (2)/(4-sqrt6) simplifies to 2.
Sure! To subtract fractions, we need to have a common denominator.
Step 1: Find the common denominator.
The denominators in your expression are (4 + √6) and (4 - √6). To get a common denominator, we can multiply these denominators together.
Common denominator = (4 + √6)(4 - √6)
Step 2: Simplify the expression.
Now that we have a common denominator, we can rewrite the expression:
(2)/(4 + √6) - (2)/(4 - √6) = [(2)(4 - √6)] / [(4 + √6)(4 - √6)] - [(2)(4 + √6)] / [(4 + √6)(4 - √6)]
Step 3: Simplify further.
Let's simplify both numerator terms separately:
[(2)(4 - √6)] = 8 - 2√6
[(2)(4 + √6)] = 8 + 2√6
Now rewrite the expression:
(8 - 2√6) / [(4 + √6)(4 - √6)] - (8 + 2√6) / [(4 + √6)(4 - √6)]
Step 4: Combine the terms.
Since we have the same denominator [(4 + √6)(4 - √6)], we can subtract the fractions:
[(8 - 2√6) - (8 + 2√6)] / [(4 + √6)(4 - √6)]
Simplifying the numerator:
(8 - 2√6) - (8 + 2√6) = 8 - 2√6 - 8 - 2√6 = -4√6
Now rewrite the expression:
(-4√6) / [(4 + √6)(4 - √6)]
Step 5: Simplify the expression further.
We can simplify the denominator:
[(4 + √6)(4 - √6)] = 4^2 - (√6)^2 = 16 - 6 = 10
So the final expression is:
-4√6 / 10
Step 6: Simplify the fraction.
We can simplify the fraction further by reducing the numerator and denominator by their greatest common divisor:
-4√6 / 10 = (-2√6) / 5
Therefore, the subtraction of the given fractions is (-2√6) / 5.