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Difficulty: HardEvaluating Algebraic Expressions

Find the value of the following expression for x=−3x = -3, y=−2y = -2, and z=12z = \frac{1}{2}:

x2y−y3x−y+4z \frac{x^2 y - y^3}{x - y} + 4z

Answer: 12

Answer

12
The correct result of 12 is obtained by substituting the given values, correctly applying exponents to negative bases, properly combining terms with double negatives, and following the standard order of operations.

Step-by-Step Solution

1
Substitute the values of x, y, and z into the algebraic expression.
(−3)2(−2)−(−2)3(−3)−(−2)+4(12)\frac{(-3)^2(-2) - (-2)^3}{(-3) - (-2)} + 4(\frac{1}{2})
Direct substitution establishes the numerical expression to be evaluated.
2
Evaluate the terms containing exponents.
(−3)2=9(-3)^2 = 9 and (−2)3=−8(-2)^3 = -8
Following the order of operations, exponents are calculated before multiplication or addition.
3
Multiply the factors in the numerator and the separate z-term.
The numerator term becomes 9(−2)=−189(-2) = -18. The z-term becomes 4(12)=24(\frac{1}{2}) = 2.
Multiplication is the next step in simplifying the expression according to the order of operations.
4
Simplify the fraction's numerator and denominator by performing the subtractions.
Numerator: −18−(−8)=−10-18 - (-8) = -10. Denominator: −3−(−2)=−1-3 - (-2) = -1.
The fraction bar acts as a grouping symbol; the numerator and denominator must be simplified completely before division.
5
Divide the simplified numerator by the simplified denominator.
−10−1=10\frac{-10}{-1} = 10
Completing the division of the rational part of the expression.
6
Add the result of the division to the remaining term.
10+2=1210 + 2 = 12
Addition is the final operation required to find the total value.

Key Concept

Evaluating complex algebraic expressions using substitution and the order of operations with negative integers and fractions.
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Topics

Evaluating Algebraic Expressions

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