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Difficulty: MediumEvaluating Algebraic Expressions

If a=−3a = -3 and b=−2b = -2, what is the value of the expression 2a2−b3(a−b)2+ab\frac{2a^2 - b^3}{(a - b)^2 + ab}?

  1. 267\frac{26}{7}Answer
  2. B
    107\frac{10}{7}
  3. C
    −107-\frac{10}{7}
  4. D
    2619\frac{26}{19}
  5. E
    2631\frac{26}{31}

Answer

267\frac{26}{7}
Substituting a=−3a = -3 and b=−2b = -2 directly into the numerator yields 2(−3)2−(−2)3=2(9)−(−8)=262(-3)^2 - (-2)^3 = 2(9) - (-8) = 26. Substituting into the denominator gives ((−3)−(−2))2+(−3)(−2)=(−1)2+6=7((-3) - (-2))^2 + (-3)(-2) = (-1)^2 + 6 = 7. Combining these results gives 267\frac{26}{7}.

Step-by-Step Solution

1
Substitute a=−3a = -3 and b=−2b = -2 into the numerator 2a2−b32a^2 - b^3.
2(−3)2−(−2)3=2(9)−(−8)=18+8=262(-3)^2 - (-2)^3 = 2(9) - (-8) = 18 + 8 = 26
Squaring −3-3 gives 99, and cubing −2-2 gives −8-8. Subtracting −8-8 is equivalent to adding 88.
2
Substitute a=−3a = -3 and b=−2b = -2 into the denominator (a−b)2+ab(a - b)^2 + ab.
((−3)−(−2))2+(−3)(−2)=(−3+2)2+6=(−1)2+6=1+6=7((-3) - (-2))^2 + (-3)(-2) = (-3 + 2)^2 + 6 = (-1)^2 + 6 = 1 + 6 = 7
Subtracting a negative number becomes addition, and squaring −1-1 yields 11.
3
Form the fraction by dividing the numerator by the denominator.
267\frac{26}{7}
The evaluated numerator is 2626 and the evaluated denominator is 77.

Key Concept

Order of operations and handling negative signs when evaluating algebraic expressions involving powers and parentheses.
Estimated Time:1m 0s
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Topics

Evaluating Algebraic Expressions

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