How would I simplify (1/2sqrtx^(-1/2))/(sqrt2x+9?
eh? Do you mean
1/2 (√x)^(-1/2)
-----------------
√(2x+9)
Kind of odd, having a root of a root
I'm sorry no, all of it is multiplied (just ignore the dividing slash before square root2x+9, meant to be parentheses to multiply).
Well, if you mean
1/2 (√x)^(-1/2) √(2x+9)
There's not much to do. If you make it all a 4th-root, then you have
1/2 ∜((2x+9)^2 / x)
I still don't think that's what you meant, but ...
To simplify the expression (1/2sqrtx^(-1/2))/(sqrt2x+9), we can start by simplifying each term separately and then applying the rules of fractions.
Let's break it down step by step:
Step 1: Simplify the denominator
The denominator is sqrt(2x) + 9. We don't have any direct way to simplify it further, so we'll leave it as it is for now.
Step 2: Simplify the numerator
The numerator is (1/2)sqrt(x^(-1/2)). We can simplify this by using the properties of exponents and radicals.
The exponent -1/2 indicates that x^(-1/2) is the reciprocal of the square root of x. So we can rewrite x^(-1/2) as 1/sqrt(x).
Therefore, the numerator becomes (1/2)sqrt(1/sqrt(x)).
Step 3: Combine the numerator and denominator
Now that we have simplified the numerator and denominator separately, let's combine them.
The expression becomes (1/2)sqrt(1/sqrt(x))/(sqrt(2x) + 9).
Step 4: Rationalize the denominator
To rationalize the denominator, we need to eliminate any irrational numbers (such as square roots) from the denominator.
Multiply both the numerator and the denominator by the conjugate of the denominator, which is sqrt(2x) - 9.
The expression becomes [(1/2)sqrt(1/sqrt(x))(sqrt(2x) - 9)] / [(sqrt(2x) + 9)(sqrt(2x) - 9)].
Step 5: Simplify further
Now, let's simplify the expression further by expanding and simplifying.
The expression becomes [(1/2)sqrt(2x - 18x^(1/2)) - 9sqrt(1/sqrt(x))] / [2x - 81].
That's the simplified form of the expression (1/2sqrtx^(-1/2))/(sqrt2x+9).