Math Problem Statement

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

Solution

To simplify the expression x21+x2\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 1+x2\sqrt{1 + x^2}. Here's how the simplification works:

x21+x2×1+x21+x2=x21+x21+x2\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+x21 + x^2, which simplifies the expression to:

x21+x21+x2\frac{x^2 \sqrt{1 + x^2}}{1 + x^2}

This is the simplified expression with a denominator of 1+x21 + 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 1x2\sqrt{1 - x^2} instead?
  2. What happens if we substitute specific values for xx in the simplified expression?
  3. How can we further manipulate this expression for xx \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=0x = 0?

Tip: Simplifying expressions often involves multiplying by clever forms of 1 (like 1+x21+x2\frac{\sqrt{1 + x^2}}{\sqrt{1 + x^2}}) to achieve the desired form.

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplification
Rationalization of Denominators
Radicals

Formulas

Rationalizing a denominator using sqrt(1 + x^2)

Theorems

Rationalization theorem for expressions with radicals

Suitable Grade Level

Grades 10-12