simplify √(23+√408) -√(23-√408) ?
Let 23=x
408=y
N= final answer
then
√(23+√408) - √(23-√408) becomes
√(x+√y) - √(x-√y)
Equating it to N
√(x+√y) - √(x-√y)=N
Solving this,
N²= 2x-2√(x²-y)
Subsituting x=23 and y=408
N²=2(23)-2√(23²-408)
=46-2√(529-408)
=46-2√121
=46-2(11)
=46-22
=24
N²=24
N=2√6.
I hope it helps.
LOL
a^2 - b^2 = (a+b)(a-b)
so
23 - 408 = (sqrt(23) -sqrt....
To simplify the expression √(23+√408) - √(23-√408), we need to use a property of radical expressions.
Step 1: Simplify the expressions inside the square roots, if possible.
Start by simplifying √408. Since 408 is not a perfect square, we can look for factors that are perfect squares. One such factor is 4 because √4 = 2. Therefore, we can write 408 as 4 * 102.
So, √408 = √(4 * 102).
Using the product property of square roots, we can split the square root into two separate square roots, like this: √4 * √102.
Simplifying, we get 2√102.
Step 2: Substitute the simplified expressions back into the original expression.
Now, our expression becomes: √(23+2√102) - √(23-2√102).
Step 3: Further simplify the expression.
We can treat the square root of (23+2√102) and the square root of (23-2√102) as separate terms. The subtraction means that the second term will be negated.
So the expression becomes: √(23+2√102) + (-1) * √(23-2√102).
Step 4: Combine like terms.
Since the square roots have different radicands (numbers inside the roots), there are no like terms to combine in this case.
Therefore, the simplified expression is: √(23+2√102) - √(23-2√102).