Simply the expression x^2/sqrt(1+x^2) so that only a 1 remains in the denominator

To simplify the expression \(\frac{x^2}{\sqrt{1 + x^2}}\) such that only a 1 remains in the denominator, we can multiply both the numerator and denominator by \(\sqrt{1 + x^2}\). Here's how the simplification works:

\[
\frac{x^2}{\sqrt{1 + x^2}} \times \frac{\sqrt{1 + x^2}}{\sqrt{1 + x^2}} = \frac{x^2 \sqrt{1 + x^2}}{1 + x^2}
\]

Now, the denominator becomes \(1 + x^2\), which simplifies the expression to:

\[
\frac{x^2 \sqrt{1 + x^2}}{1 + x^2}
\]

This is the simplified expression with a denominator of \(1 + x^2\).

Would you like more details or explanations on this step? Here are 5 related questions you might explore:

1. How would the result change if the denominator was \(\sqrt{1 - x^2}\) instead?
2. What happens if we substitute specific values for \(x\) in the simplified expression?
3. How can we further manipulate this expression for \(x \to \infty\)?
4. What are the possible interpretations of this expression in calculus?
5. Can you express the original function as a Taylor series around \(x = 0\)?

**Tip:** Simplifying expressions often involves multiplying by clever forms of 1 (like \(\frac{\sqrt{1 + x^2}}{\sqrt{1 + x^2}}\)) to achieve the desired form.

Learn how to simplify the expression x^2/sqrt(1 + x^2) by multiplying both numerator and denominator by sqrt(1 + x^2). This solution uses key concepts like rationalizing denominators and simplifying radicals, suitable for high school students in Grades 10-12.

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