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Difficulty: HardEquivalent Algebraic Expressions

For all positive real numbers aa and bb such that a≠ba \neq b, which of the following is equivalent to the expression a3/2−b3/2a−b\frac{a^{3/2} - b^{3/2}}{\sqrt{a} - \sqrt{b}}?

  1. a+ab+ba + \sqrt{ab} + bAnswer
  2. B
    a+ba + b
  3. C
    a−ab+ba - \sqrt{ab} + b
  4. D
    a+2ab+ba + 2\sqrt{ab} + b

Answer

a+ab+ba + \sqrt{ab} + b
The correct answer is obtained by expressing the numerator as a difference of cubes: (a)3−(b)3(\sqrt{a})^3 - (\sqrt{b})^3. Factoring this expression gives (a−b)(a+ab+b)(\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b). Since a≠ba \neq b, dividing by the denominator a−b\sqrt{a} - \sqrt{b} simplifies the expression to a+ab+ba + \sqrt{ab} + b.

Step-by-Step Solution

1
Rewrite the terms in the numerator using square roots to reveal a difference of cubes pattern.
a3/2=(a)3a^{3/2} = (\sqrt{a})^3 and b3/2=(b)3b^{3/2} = (\sqrt{b})^3, so the numerator is (a)3−(b)3(\sqrt{a})^3 - (\sqrt{b})^3.
This allows us to factor the numerator using the algebraic identity for the difference of two cubes.
2
Factor the numerator using the difference of cubes formula: u3−v3=(u−v)(u2+uv+v2)u^3 - v^3 = (u - v)(u^2 + uv + v^2).
(a)3−(b)3=(a−b)(a+ab+b)(\sqrt{a})^3 - (\sqrt{b})^3 = (\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b) where u=au = \sqrt{a} and v=bv = \sqrt{b}.
Factoring allows us to identify common factors shared with the denominator.
3
Substitute the factored expression back into the fraction and cancel the common factor of a−b\sqrt{a} - \sqrt{b}.
(a−b)(a+ab+b)a−b=a+ab+b\frac{(\sqrt{a} - \sqrt{b})(a + \sqrt{ab} + b)}{\sqrt{a} - \sqrt{b}} = a + \sqrt{ab} + b.
Since a≠ba \neq b, a−b≠0\sqrt{a} - \sqrt{b} \neq 0, which makes it mathematically valid to divide by this term.

Key Concept

Equivalent algebraic expressions involving fractional exponents and difference of cubes factoring

Alternative Method

Let a=4a = 4 and b=1b = 1. Substitute these values into the original expression: 43/2−13/24−1=8−12−1=7\frac{4^{3/2} - 1^{3/2}}{\sqrt{4} - \sqrt{1}} = \frac{8 - 1}{2 - 1} = 7. Now substitute these same values into each option to see which one evaluates to 7. The correct option evaluates to 4+4(1)+1=74 + \sqrt{4(1)} + 1 = 7.
Estimated Time:2m 0s
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Topics

Equivalent Algebraic Expressions

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