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Difficulty: MediumEvaluating Algebraic Expressions

What is the evaluated result of the algebraic expression a2−4ab+b3a^2 - 4ab + b^3 when a=−5a = -5 and b=−2b = -2?

  1. A
    57
  2. B
    -7
  3. -23Answer
  4. D
    -57

Answer

The correct evaluated result is -23.
The correct answer is found by substituting a=−5a = -5 and b=−2b = -2 into the expression and strictly following the order of operations. Squaring −5-5 results in 2525. The product of −4(−5)(−2)-4(-5)(-2) is −40-40. Cubing −2-2 results in −8-8. Combining these gives 25−40−825 - 40 - 8, which equals −23-23.

Step-by-Step Solution

1
Substitute the given values a=−5a = -5 and b=−2b = -2 into the expression.
(−5)2−4(−5)(−2)+(−2)3(-5)^2 - 4(-5)(-2) + (-2)^3
Direct substitution is the first step to evaluating an algebraic expression.
2
Evaluate the exponential terms.
25−4(−5)(−2)+(−8)25 - 4(-5)(-2) + (-8)
According to the order of operations (PEMDAS), exponents must be calculated before multiplication. Squaring a negative number gives a positive result ((−5)2=25(-5)^2 = 25), and cubing a negative number gives a negative result ((−2)3=−8(-2)^3 = -8).
3
Calculate the multiplication in the middle term.
25−40−825 - 40 - 8
Multiply the three factors: −4×−5×−2-4 \times -5 \times -2. Two negatives make a positive (2020), and multiplying by the third negative makes it negative (−40-40).
4
Combine the simplified terms using addition and subtraction from left to right.
−15−8=−23-15 - 8 = -23
Final arithmetic computation yields the fully evaluated numerical value.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:1m 0s
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Topics

Evaluating Algebraic Expressions

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