Question

Упростить (x^2+3)*(9+x^4+6x^2)^(-1/2)

Answer 1

$(x^{2} +3)*(9+x^{4}+6x^{2})^{-1/2} = \frac{x^{2}+3}{\sqrt{9+x^{4}+6x^{2}}} = \frac{x^{2}+3}{\sqrt{(3+x^{2})^{2}}} = \frac{x^{2}+3}{x^{2}+3} =1$

Answer 2

To simplify the expression, we need to rewrite the denominator in a more manageable form using the properties of exponents.

Let's start by factoring the denominator:

Denominator: (9 + x^4 + 6x^2)

Now, we notice that this is a perfect square trinomial, which can be factored as follows:

Denominator: [(x^4 + 6x^2) + 9]

Now, let's focus on the squared part within the bracket:

x^4 + 6x^2

We can factor out x^2:

x^4 + 6x^2 = x^2(x^2 + 6)

Now the denominator becomes:

Denominator: [(x^2)(x^2 + 6) + 9]

Now we can rewrite the entire expression:

(x^2 + 3) * [(x^2)(x^2 + 6) + 9]^(-1/2)

Now, we'll simplify the expression in the square brackets by raising it to the power of -1/2 (which is the same as taking the square root):

[(x^2)(x^2 + 6) + 9]^(-1/2) = 1 / √[(x^2)(x^2 + 6) + 9]

So, the simplified expression is:

(x^2 + 3) / √[(x^2)(x^2 + 6) + 9]