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Difficulty: HardEquivalent Algebraic Expressions

For all x>8x > 8, the expression x8/3−8x5/3x4/3−4x2/3⋅x1/3+2x4/3+2x+4x2/3\frac{x^{8/3} - 8x^{5/3}}{x^{4/3} - 4x^{2/3}} \cdot \frac{x^{1/3} + 2}{x^{4/3} + 2x + 4x^{2/3}} is equivalent to xax^a, where aa is a constant. What is the value of 1a\frac{1}{a}?

Answer: 3

Answer

The correct answer is 3.
Factoring the numerator and denominator of the first fraction yields x5/3(x1/3−2)(x2/3+2x1/3+4)x2/3(x1/3−2)(x1/3+2)\frac{x^{5/3}(x^{1/3}-2)(x^{2/3}+2x^{1/3}+4)}{x^{2/3}(x^{1/3}-2)(x^{1/3}+2)}, which simplifies to x(x2/3+2x1/3+4)x1/3+2\frac{x(x^{2/3}+2x^{1/3}+4)}{x^{1/3}+2}. Factoring the denominator of the second fraction yields x1/3+2x2/3(x2/3+2x1/3+4)\frac{x^{1/3}+2}{x^{2/3}(x^{2/3}+2x^{1/3}+4)}. Multiplying these two simplified expressions cancels the common terms (x1/3+2)(x^{1/3}+2) and (x2/3+2x1/3+4)(x^{2/3}+2x^{1/3}+4), leaving xx2/3=x1/3\frac{x}{x^{2/3}} = x^{1/3}. Therefore, a=13a = \frac{1}{3}, and the value of the reciprocal 1a\frac{1}{a} is 33.

Step-by-Step Solution

1
Factor the numerator and the denominator of the first fraction.
The first fraction becomes x(x2/3+2x1/3+4)x1/3+2\frac{x(x^{2/3} + 2x^{1/3} + 4)}{x^{1/3} + 2}.
Factoring out x5/3x^{5/3} from the numerator gives x5/3(x−8)x^{5/3}(x-8), and factoring out x2/3x^{2/3} from the denominator gives x2/3(x2/3−4)x^{2/3}(x^{2/3}-4). Using the difference of cubes x−8=(x1/3−2)(x2/3+2x1/3+4)x - 8 = (x^{1/3} - 2)(x^{2/3} + 2x^{1/3} + 4) and the difference of squares x2/3−4=(x1/3−2)(x1/3+2)x^{2/3} - 4 = (x^{1/3} - 2)(x^{1/3} + 2), we can cancel the common factor (x1/3−2)(x^{1/3} - 2).
2
Factor the denominator of the second fraction.
The second fraction becomes x1/3+2x2/3(x2/3+2x1/3+4)\frac{x^{1/3} + 2}{x^{2/3}(x^{2/3} + 2x^{1/3} + 4)}.
Factoring out x2/3x^{2/3} from the expression x4/3+2x+4x2/3x^{4/3} + 2x + 4x^{2/3} reveals a common quadratic-like term (x2/3+2x1/3+4)(x^{2/3} + 2x^{1/3} + 4) that can be used for cancellation.
3
Multiply the two rational expressions together and simplify.
x1/3x^{1/3}
Multiplying the simplified fractions allows us to cancel the common binomial term (x1/3+2)(x^{1/3} + 2) and the trinomial term (x2/3+2x1/3+4)(x^{2/3} + 2x^{1/3} + 4), leaving xx2/3=x1−2/3=x1/3\frac{x}{x^{2/3}} = x^{1 - 2/3} = x^{1/3}.
4
Find the value of 1a\frac{1}{a}.
3
Since the expression is equivalent to xax^a, we identify a=13a = \frac{1}{3}. Taking the reciprocal of aa gives 11/3=3\frac{1}{1/3} = 3.

Key Concept

Equivalent Algebraic Expressions
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Equivalent Algebraic Expressions

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