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Difficulty: HardOrder of Operations and Number Properties

For all real numbers aa and bb, which of the following expressions is equivalent to a−b[a−(b−a)]−b2a - b[a - (b - a)] - b^2?

  1. A
    aa
  2. B
    a−2ab−2b2a - 2ab - 2b^2
  3. a−2aba - 2abAnswer
  4. D
    2a2−3ab2a^2 - 3ab
  5. E
    a−2ab+2b2a - 2ab + 2b^2

Answer

The expression a−2aba - 2ab
To simplify the expression, we begin inside the innermost grouping symbols (the parentheses) and work outward. First, we distribute the negative sign to simplify the term inside the brackets: a−(b−a)=a−b+a=2a−ba - (b - a) = a - b + a = 2a - b. Next, we distribute the −b-b term outside the brackets to get −b(2a−b)=−2ab+b2-b(2a - b) = -2ab + b^2. Substituting this back into the main expression gives a−2ab+b2−b2a - 2ab + b^2 - b^2. Combining the like terms b2−b2=0b^2 - b^2 = 0 simplifies the expression to a−2aba - 2ab.

Step-by-Step Solution

1
Simplify the expression inside the brackets by distributing the negative sign across the parentheses.
a−(b−a)=a−b+a=2a−ba - (b - a) = a - b + a = 2a - b
According to the order of operations, terms inside parentheses must be simplified first.
2
Substitute the simplified bracket expression back into the main expression.
a−b[2a−b]−b2a - b[2a - b] - b^2
This sets up the expression for the next operation, which is multiplication.
3
Distribute the term −b-b across the bracketed expression.
−b(2a−b)=−2ab+b2-b(2a - b) = -2ab + b^2
Multiplication must be performed before subtraction. Multiplying two negative signs yields a positive term.
4
Combine the distributed terms back into the full expression and simplify like terms.
a−2ab+b2−b2=a−2aba - 2ab + b^2 - b^2 = a - 2ab
Combining the positive and negative b2b^2 terms results in 00, leaving the fully simplified expression.

Key Concept

Order of Operations and Number Properties

Alternative Method

Instead of simplifying algebraically, you can substitute simple numbers for aa and bb. Let a=2a = 2 and b=3b = 3. Evaluating the original expression: 2−3[2−(3−2)]−32=2−3[2−1]−9=2−3(1)−9=−102 - 3[2 - (3 - 2)] - 3^2 = 2 - 3[2 - 1] - 9 = 2 - 3(1) - 9 = -10. Substituting a=2a = 2 and b=3b = 3 into the correct option a−2aba - 2ab gives 2−2(2)(3)=2−12=−102 - 2(2)(3) = 2 - 12 = -10. This numerical substitution confirms the correct algebraic simplification.
Estimated Time:1m 30s
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Topics

Order of Operations and Number Properties

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