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Difficulty: MediumEvaluating Algebraic Expressions

In a mathematical model, the output variable MM is defined by the equation M=2a2−3ab+b3M = 2a^2 - 3ab + b^3. What is the computed value of MM if the input parameters are set to a=−3a = -3 and b=−2b = -2?

  1. -8Answer
  2. B
    10
  3. C
    28
  4. D
    -28

Answer

-8
The correct answer is found by substituting the values and strictly following the order of operations. First, the exponents are evaluated: (−3)2=9(-3)^2 = 9 and (−2)3=−8(-2)^3 = -8. Next, the multiplication is performed: 2(9)=182(9) = 18 and −3(−3)(−2)=−18-3(-3)(-2) = -18. Finally, combining these values gives 18−18−8=−818 - 18 - 8 = -8.

Step-by-Step Solution

1
Substitute the values a=−3a = -3 and b=−2b = -2 into the original equation M=2a2−3ab+b3M = 2a^2 - 3ab + b^3.
M=2(−3)2−3(−3)(−2)+(−2)3M = 2(-3)^2 - 3(-3)(-2) + (-2)^3
Direct substitution is the first step in evaluating any algebraic expression.
2
Evaluate the terms with exponents according to the order of operations (PEMDAS/BODMAS).
(−3)2=9(-3)^2 = 9 and (−2)3=−8(-2)^3 = -8
Exponents must be calculated before multiplication. A negative number squared is positive, while a negative number cubed remains negative.
3
Perform the multiplication for each term in the expression.
2(9)=182(9) = 18 and −3(−3)(−2)=−18-3(-3)(-2) = -18
Multiplication is performed next. For the middle term, multiplying three negative numbers results in a negative product.
4
Combine the resulting values using addition and subtraction.
18−18+(−8)=0−8=−818 - 18 + (-8) = 0 - 8 = -8
This final arithmetic operation yields the evaluated result of the expression.

Key Concept

Evaluating algebraic expressions involving exponents, negative bases, and strict adherence to the order of operations.
Estimated Time:1m 0s
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Topics

Evaluating Algebraic Expressions

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