Math Problem Statement

Simplify the expression -((x^2 + 1)^(1/2) + x^2 * (x^2 + 1)^(-1/2)) / x^2.

Solution

sympy import symbols, sqrt, simplify

Define the variables and expression

x = symbols('x') expression = (-(x2 + 1)(1/2) + x2 * (x2 + 1)(-1/2)) / x2

Simplify the expression

simplified_expression = simplify(expression) simplified_expressionThe simplified expression is:

(x2+1)1/2(x2+1)1/2x2(x^2 + 1)^{-1/2} - \frac{(x^2 + 1)^{1/2}}{x^2}

Would you like further details on the steps involved in the simplification or any other aspect of this problem?

Here are five related questions for exploration:

  1. How does the exponentiation rule apply in simplifying this expression?
  2. What are the conditions under which this expression is defined?
  3. How can this expression be further analyzed for its limits as x0x \to 0 or xx \to \infty?
  4. How would this expression look when differentiated with respect to xx?
  5. What are some real-world applications of such expressions in mathematical physics or engineering?

Tip: Always look for opportunities to factorize or find common terms when simplifying algebraic expressions.

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Math Problem Analysis

Mathematical Concepts

Algebra
Exponentiation
Simplification

Formulas

a^m * a^n = a^(m+n)
(a^m)^n = a^(m*n)
(a + b) / c = a/c + b/c

Theorems

Laws of Exponents

Suitable Grade Level

Grades 11-12