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Zorluk: Çok zorEquivalent 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 + 1Cevap
  4. D
    x−1x - 1

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Simplifying rational expressions with fractional exponents by substitution and factoring.
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Konular

Equivalent Algebraic Expressions

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