Simplify x^2/((x+4)(sqrt(x^2+9)-3)
To simplify the expression x^2/((x+4)(sqrt(x^2+9)-3), we can start by factoring the denominator.
(x+4)(sqrt(x^2+9)-3) = (x+4)(sqrt(x^2+9)-3)
= (x+4)(sqrt(x^2+9)-3)
Next, we can look for common factors between the numerator and the denominator. In this case, we see that x^2 is a common factor.
x^2/((x+4)(sqrt(x^2+9)-3) = (x^2/(x+4))(1/(sqrt(x^2+9)-3))
Now, we can simplify the expression by canceling out the common factors.
(x^2/(x+4))(1/(sqrt(x^2+9)-3)) = (x^2/(x+4))/((sqrt(x^2+9)-3))
Therefore, the simplified expression is x^2/(x+4)/((sqrt(x^2+9)-3)).
To simplify the expression x^2/((x+4)(sqrt(x^2+9)-3), we can start by simplifying the denominator.
The denominator contains two terms: (x+4) and (sqrt(x^2+9)-3).
Let's simplify the second term, (sqrt(x^2+9)-3), first.
We can simplify this term by rationalizing the denominator, which means getting rid of the square root in the denominator.
To do that, we multiply the numerator and denominator of the term by the conjugate of the denominator, which is (sqrt(x^2+9)+3).
So, we now have (sqrt(x^2+9)-3) * (sqrt(x^2+9)+3) as the new denominator.
Using the difference of squares pattern, (a^2 - b^2) = (a + b)(a - b), we can simplify the denominator further.
In this case, a = sqrt(x^2+9) and b = 3, so (sqrt(x^2+9))^2 - 3^2 = (x^2 + 9 - 9) = x^2.
Therefore, the denominator simplifies to x^2.
Now, we can rewrite the initial expression as x^2/(x+4)*x^2.
To further simplify, we can multiply the x^2 terms together:
x^2 * x^2 = x^4
Finally, the simplified expression is x^4/(x+4).
So, x^2/((x+4)(sqrt(x^2+9)-3) simplifies to x^4/(x+4).
To simplify the expression x^2/((x+4)(√(x^2+9)-3)), we can start by factoring the denominator.
The denominator can be factored using the difference of squares formula:
x^2 + 9 = (x + 3)(x - 3)
So, the expression can be rewritten as:
x^2/((x+4)(√(x^2+9)-3)) = x^2/((x+4)(√(x + 3)(x - 3))-3))
Now, let's simplify the expression further.
Next, we will simplify the denominator by multiplying the conjugate of the second term (√(x^2 + 9) - 3). The conjugate of √(x^2 + 9) - 3 is √(x^2 + 9) + 3.
When we multiply the numerator and denominator by the conjugate, the denominator will become a difference of squares expression.
So, the expression becomes:
x^2/[ (x + 4)(√(x + 3)(x - 3)) - 3(√(x + 3) + 3)]
Expanding the denominator, we get:
x^2/[x(√(x + 3)(x - 3)) - 3(√(x + 3)(x - 3)) + 4√(x + 3)(x - 3) - 9√(x + 3) - 12]
Further simplifying:
x^2/[x(√(x + 3)(x - 3)) - 3(√(x + 3)(x - 3)) + 4√(x + 3)(x - 3) - 9√(x + 3) - 12]
= x^2/[x√(x + 3)(x - 3) - 3√(x + 3)(x - 3) + 4√(x + 3)(x - 3) - 9√(x + 3) - 12]
Now, we can combine the terms on the denominator:
x^2/[x√(x + 3)(x - 3) + √(x + 3)(x - 3) - 9√(x + 3) - 12]
Factoring out the common factor √(x + 3)(x - 3) from the first two terms in the denominator, we get:
x^2/[(x - 3)(x√(x + 3) + √(x + 3)) - 9√(x + 3) - 12]
Finally, we can simplify the expression to its simplified form:
x^2/[(x - 3)(x + 1)√(x + 3) - 9√(x + 3) - 12]
And that is the simplified form of the expression x^2/((x+4)(√(x^2+9)-3)).