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Difficulty: HardEvaluating Algebraic Expressions

If x=−2x = -2, y=12y = \frac{1}{2}, and z=−3z = -3, what is the value of the algebraic expression x3y−2−4z(x+2y)2−z\frac{x^3 y^{-2} - 4z}{(x + 2y)^2 - z}?

  1. −5-5Answer
  2. B
    −10-10
  3. C
    −52-\frac{5}{2}
  4. D
    52\frac{5}{2}
  5. E
    −11-11

Answer

The value of the expression is −5-5.
Substituting x=−2x = -2, y=12y = \frac{1}{2}, and z=−3z = -3 yields a numerator of (−2)3(12)−2−4(−3)=(−8)(4)+12=−20(-2)^3 \left(\frac{1}{2}\right)^{-2} - 4(-3) = (-8)(4) + 12 = -20, and a denominator of (−2+2(12))2−(−3)=(−1)2+3=4\left(-2 + 2\left(\frac{1}{2}\right)\right)^2 - (-3) = (-1)^2 + 3 = 4. Dividing −20-20 by 44 gives the final answer −5-5.

Step-by-Step Solution

1
Evaluate the terms in the numerator: x3y−2−4zx^3 y^{-2} - 4z
Since x=−2x = -2, x3=(−2)3=−8x^3 = (-2)^3 = -8. Since y=12y = \frac{1}{2}, y−2=(12)−2=22=4y^{-2} = \left(\frac{1}{2}\right)^{-2} = 2^2 = 4. So x3y−2=(−8)(4)=−32x^3 y^{-2} = (-8)(4) = -32. Also, −4z=−4(−3)=12-4z = -4(-3) = 12. The numerator simplifies to −32+12=−20-32 + 12 = -20.
Negative exponents indicate reciprocals, and cubing a negative base yields a negative result.
2
Evaluate the terms in the denominator: (x+2y)2−z(x + 2y)^2 - z
Inside the parentheses, x+2y=−2+2(12)=−2+1=−1x + 2y = -2 + 2\left(\frac{1}{2}\right) = -2 + 1 = -1. Squaring this gives (−1)2=1(-1)^2 = 1. Subtracting zz gives 1−(−3)=1+3=41 - (-3) = 1 + 3 = 4.
Operations inside parentheses must be calculated before applying the exponent, and subtracting a negative integer is equivalent to adding its positive.
3
Divide the numerator by the denominator
−204=−5.\frac{-20}{4} = -5.
Dividing a negative integer by a positive integer produces a negative quotient.

Key Concept

Evaluating algebraic expressions involving negative exponents, integer substitutions, and order of operations.
Estimated Time:1m 30s
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Topics

Evaluating Algebraic Expressions

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