Examkin

Question

Difficulty: Very hardEquivalent Algebraic Expressions

For all x>1x > 1, which of the following is equivalent to the expression x2−xx−x−x\frac{x^2 - \sqrt{x}}{x - \sqrt{x}} - \sqrt{x}?

  1. A
    x−2x+1x - 2\sqrt{x} + 1
  2. B
    x−x−1x - \sqrt{x} - 1
  3. x+1x + 1Answer
  4. D
    x−1x - 1

Answer

The expression is equivalent to x+1x + 1.
The expression can be simplified by substituting u=xu = \sqrt{x}, which gives x=u2x = u^2 and x2=u4x^2 = u^4. Substituting these into the original expression yields u4−uu2−u−u\frac{u^4 - u}{u^2 - u} - u. Factoring out uu from the numerator and denominator gives u(u3−1)u(u−1)−u=u3−1u−1−u\frac{u(u^3 - 1)}{u(u - 1)} - u = \frac{u^3 - 1}{u - 1} - u. Factoring the difference of cubes in the numerator as (u−1)(u2+u+1)(u - 1)(u^2 + u + 1) and canceling the common factor of u−1u - 1 leaves u2+u+1−u=u2+1u^2 + u + 1 - u = u^2 + 1. Substituting back x=u2x = u^2 yields the equivalent expression x+1x + 1.

Step-by-Step Solution

1
Substitute u=xu = \sqrt{x} into the expression, which implies x=u2x = u^2 and x2=u4x^2 = u^4.
The expression becomes u4−uu2−u−u\frac{u^4 - u}{u^2 - u} - u.
Using a substitution simplifies the fractional exponents and makes the polynomial structure easier to recognize.
2
Factor out uu from both the numerator and the denominator of the fraction.
u(u3−1)u(u−1)−u=u3−1u−1−u\frac{u(u^3 - 1)}{u(u - 1)} - u = \frac{u^3 - 1}{u - 1} - u.
Since x>1x > 1, we have u>1u > 1, which means u≠0u \neq 0. Therefore, we can cancel the common factor uu from the numerator and denominator.
3
Factor the difference of cubes in the numerator: u3−1=(u−1)(u2+u+1)u^3 - 1 = (u - 1)(u^2 + u + 1).
(u−1)(u2+u+1)u−1−u\frac{(u - 1)(u^2 + u + 1)}{u - 1} - u.
Factoring the numerator allows us to simplify the rational expression by canceling the common binomial factor in the denominator.
4
Cancel the common factor u−1u - 1 and simplify the remaining terms.
(u2+u+1)−u=u2+1(u^2 + u + 1) - u = u^2 + 1.
Since u>1u > 1, we have u−1≠0u - 1 \neq 0, allowing us to divide out u−1u - 1. Subtracting uu from u2+u+1u^2 + u + 1 leaves u2+1u^2 + 1.
5
Substitute xx back in place of u2u^2.
x+1x + 1.
Converting the simplified expression back to the original variable gives the final equivalent algebraic expression.

Key Concept

Simplifying rational expressions with fractional exponents by substitution and factoring.
Rate this question

Topics

Equivalent Algebraic Expressions

More questions from this topic