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Simplifying the Expression: $(x^2+1)^{1/2} + 2(x^2+1)^{-1/2}$
In this article, we will simplify the given expression: $(x^2+1)^{1/2} + 2(x^2+1)^{-1/2}$. To do this, we will use some algebraic manipulations and properties of exponents.
Simplifying the First Term
The first term is $(x^2+1)^{1/2}$. This is a square root of the sum of $x^2$ and 1. We can rewrite this as:
$(x^2+1)^{1/2} = \sqrt{x^2+1}$
Simplifying the Second Term
The second term is $2(x^2+1)^{-1/2}$. This is a multiple of the reciprocal of the square root of the sum of $x^2$ and 1. We can rewrite this as:
$2(x^2+1)^{-1/2} = \frac{2}{\sqrt{x^2+1}}$
Combining the Terms
Now that we have simplified each term, we can combine them:
$(x^2+1)^{1/2} + 2(x^2+1)^{-1/2} = \sqrt{x^2+1} + \frac{2}{\sqrt{x^2+1}}$
Rationalizing the Denominator
To simplify the expression further, we can rationalize the denominator of the second term:
$\frac{2}{\sqrt{x^2+1}} = \frac{2\sqrt{x^2+1}}{\sqrt{x^2+1}\sqrt{x^2+1}} = \frac{2\sqrt{x^2+1}}{x^2+1}$
Final Simplification
Therefore, the simplified expression is:
$(x^2+1)^{1/2} + 2(x^2+1)^{-1/2} = \sqrt{x^2+1} + \frac{2\sqrt{x^2+1}}{x^2+1}$
And that's it! We have successfully simplified the given expression.
