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Difficulty: HardEquivalent Algebraic Expressions

For all x>0x > 0, which of the following expressions is equivalent to x2−8x4/3+2x2/3+4+x−4x1/3x2/3+2x1/3\frac{x^2 - 8}{x^{4/3} + 2x^{2/3} + 4} + \frac{x - 4x^{1/3}}{x^{2/3} + 2x^{1/3}}?

  1. A
    x2/3+x1/3+4x^{2/3} + x^{1/3} + 4
  2. B
    x4/3+x2/3−4x^{4/3} + x^{2/3} - 4
  3. x2/3+x1/3−4x^{2/3} + x^{1/3} - 4Answer
  4. D
    x2/3+x1/3−6x^{2/3} + x^{1/3} - 6

Answer

The expression x2/3+x1/3−4x^{2/3} + x^{1/3} - 4
The correct answer shows the sum of the simplified terms, which is x2/3+x1/3−4x^{2/3} + x^{1/3} - 4. First, the numerator of the first term, x2−8x^2 - 8, can be written as a difference of cubes: (x2/3)3−23=(x2/3−2)(x4/3+2x2/3+4)(x^{2/3})^3 - 2^3 = (x^{2/3} - 2)(x^{4/3} + 2x^{2/3} + 4). Dividing by the denominator leaves x2/3−2x^{2/3} - 2. Second, the second term can be factored by extracting x1/3x^{1/3} from both the numerator and denominator, leaving x2/3−4x1/3+2\frac{x^{2/3} - 4}{x^{1/3} + 2}. Factoring the numerator as a difference of squares, (x1/3−2)(x1/3+2)(x^{1/3} - 2)(x^{1/3} + 2), and dividing by the denominator leaves x1/3−2x^{1/3} - 2. Adding the two simplified parts, (x2/3−2)+(x1/3−2)(x^{2/3} - 2) + (x^{1/3} - 2), gives x2/3+x1/3−4x^{2/3} + x^{1/3} - 4.

Step-by-Step Solution

1
Simplify the first term, x2−8x4/3+2x2/3+4\frac{x^2 - 8}{x^{4/3} + 2x^{2/3} + 4}
x2/3−2x^{2/3} - 2
Rewrite x2−8x^2 - 8 as (x2/3)3−23(x^{2/3})^3 - 2^3 and expand using the difference of cubes identity: a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2), where a=x2/3a = x^{2/3} and b=2b = 2. The term x4/3+2x2/3+4x^{4/3} + 2x^{2/3} + 4 in the numerator and denominator cancels out.
2
Simplify the second term, x−4x1/3x2/3+2x1/3\frac{x - 4x^{1/3}}{x^{2/3} + 2x^{1/3}}
x1/3−2x^{1/3} - 2
Factor out x1/3x^{1/3} from both the numerator and denominator to get x2/3−4x1/3+2\frac{x^{2/3} - 4}{x^{1/3} + 2}. Then rewrite x2/3−4x^{2/3} - 4 as (x1/3)2−22(x^{1/3})^2 - 2^2 and expand using the difference of squares identity: a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b), where a=x1/3a = x^{1/3} and b=2b = 2. The term x1/3+2x^{1/3} + 2 in the numerator and denominator cancels out.
3
Sum the two simplified terms
x2/3+x1/3−4x^{2/3} + x^{1/3} - 4
Add the two simplified expressions: (x2/3−2)+(x1/3−2)=x2/3+x1/3−4(x^{2/3} - 2) + (x^{1/3} - 2) = x^{2/3} + x^{1/3} - 4.

Key Concept

Simplification of rational expressions involving fractional exponents, difference of cubes, and difference of squares.
Estimated Time:2m 0s
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Topics

Equivalent Algebraic Expressions

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