Simplify and show work:
[(X^(2)+3)^(1/3)](-1)(x^(-2))+[1/3(x^(2)+3)^(-2/3)](2x)(x^(-1))
My answer key says that it's this but I have no idea how to get to it. I've tried 4 times already:
[-(x^(2)+9)]/[(3x^(2))(x^(2)+3))^(2/3)]
you cannot simplify with negatives
[(X^(2)+3)^(1/3)](-1)(x^(-2))+[1/3(x^(2)+3)^(-2/3)](2x)(x^(-1))
so the real question is
[(X^2)+3)^(1/3)](-1)(x^(-2))+[1/3(x^(2)+3)^(-2/3)](2x)(x^(-1))
i hope this helped
[(X^(2)+3)^(1/3)](-1)(x^(-2))+[1/3(x^(2)+3)^(-2/3)](2x)(x^(-1))
= [ ( x^2 + 3)^(1/3) ] ( -1/x^2 ) + [ (1/3) /(x^2 + 3)(2/3) ](2x/x)
I entered it exactly the way you typed it, and did not get that answer.
Check your brackets
http://www.wolframalpha.com/input/?i=simplify+%5B(X%5E(2)%2B3)%5E(1%2F3)%5D(-1)(x%5E(-2))%2B%5B1%2F3(x%5E(2)%2B3)%5E(-2%2F3)%5D(2x)(x%5E(-1))
Go to Reiny's url and tweak the input until it agrees with what you intended. It's probably just a combination of typos.
To simplify the given expression, let's break it down into smaller parts and simplify each part individually.
Let's start with the first term: [(X^(2)+3)^(1/3)](-1)(x^(-2)).
Step 1: Simplify the first term [(X^(2)+3)^(1/3)].
To simplify a cube root expression, we can rewrite it as a fractional exponent. Thus, [(X^(2)+3)^(1/3)] can be written as (X^(2)+3)^(1/3) = (X^(2)+3)^(1/3 * 3/1) = (X^(2)+3)^(3/3).
Step 2: Simplify the exponent (3/3). The numerator (3) and the denominator (3) in the exponent cancel each other out, leaving us with (X^(2)+3).
So, the expression simplifies to (X^(2)+3)(-1)(x^(-2)).
Next, let's simplify the second term: [1/3(x^(2)+3)^(-2/3)](2x)(x^(-1)).
Step 1: Simplify the term [1/3(x^(2)+3)^(-2/3)].
Using the same method as before, we can rewrite the expression as (1/3)(X^(2)+3)^(-2/3).
Step 2: Simplify the [(X^(2)+3)^(-2/3)].
To simplify a negative fractional exponent, we can rewrite the expression as the reciprocal of the positive exponent. Thus, (X^(2)+3)^(-2/3) = 1 / (X^(2)+3)^(2/3).
So, the expression simplifies to (1/3)(1 / (X^(2)+3)^(2/3)).
Now, multiply the two terms together: (X^(2)+3)(-1)(x^(-2)) * (1/3)(1/(X^(2)+3)^(2/3)).
Step 1: Combine and simplify the numerator of the fraction.
Multiplying -1 and (X^(2)+3) gives -X^(2)-3.
Step 2: Combine the terms in the denominator.
Multiplying x^(-2) and (X^(2)+3)^(2/3) gives (X^(2)+3)^(2/3)(x^(-2)).
So, the expression [(X^(2)+3)^(1/3)](-1)(x^(-2))+[1/3(x^(2)+3)^(-2/3)](2x)(x^(-1)) simplifies to (-X^(2)-3)/(3(X^(2)+3)^(2/3)(x^2)).
However, the expression you provided as the answer is a bit different: [-(x^(2)+9)]/[(3x^(2))(x^(2)+3))^(2/3)]. It seems there might be a mistake or difference in notation. Please double-check the given answer or provide more information if necessary.