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Difficulty: HardEvaluating Algebraic Expressions
If m=−2m = -2, n=13n = \frac{1}{3}, and p=−4p = -4, what is the value of the algebraic expression below?
m3n−2−−p⋅m2m2−n−1+p\frac{m^3 n^{-2} - \sqrt{-p \cdot m^2}}{m^2 - n^{-1} + p}
  1. 763\frac{76}{3}Answer
  2. B
    7611\frac{76}{11}
  3. C
    4427\frac{44}{27}
  4. D
    −765-\frac{76}{5}
  5. E
    583\frac{58}{3}

Answer

763\frac{76}{3}
Substituting m=−2m = -2, n=13n = \frac{1}{3}, and p=−4p = -4 into the numerator gives (−2)3(13)−2−−(−4)(−2)2=−8(9)−16=−72−4=−76(-2)^3 \left(\frac{1}{3}\right)^{-2} - \sqrt{-(-4)(-2)^2} = -8(9) - \sqrt{16} = -72 - 4 = -76. Substituting into the denominator gives (−2)2−(13)−1+(−4)=4−3−4=−3(-2)^2 - \left(\frac{1}{3}\right)^{-1} + (-4) = 4 - 3 - 4 = -3. Dividing −76-76 by −3-3 gives 763\frac{76}{3}.

Step-by-Step Solution

1
Evaluate the terms in the numerator
m3=(−2)3=−8m^3 = (-2)^3 = -8, n−2=(13)−2=9n^{-2} = \left(\frac{1}{3}\right)^{-2} = 9, and −p⋅m2=−(−4)⋅(−2)2=4⋅4=4\sqrt{-p \cdot m^2} = \sqrt{-(-4) \cdot (-2)^2} = \sqrt{4 \cdot 4} = 4. Thus, Numerator =(−8)(9)−4=−72−4=−76= (-8)(9) - 4 = -72 - 4 = -76.
Simplify each term in the numerator using proper exponent rules and sign conventions.
2
Evaluate the terms in the denominator
m2=(−2)2=4m^2 = (-2)^2 = 4, n−1=(13)−1=3n^{-1} = \left(\frac{1}{3}\right)^{-1} = 3, and p=−4p = -4. Thus, Denominator =4−3+(−4)=1−4=−3= 4 - 3 + (-4) = 1 - 4 = -3.
Substitute the variable values into the denominator expression and evaluate left to right.
3
Divide the numerator by the denominator
−76−3=763.\frac{-76}{-3} = \frac{76}{3}.
Dividing two negative numbers yields a positive quotient.

Key Concept

Evaluating algebraic expressions with negative bases, fractional exponents, and order of operations
Estimated Time:2m 0s
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Topics

Evaluating Algebraic Expressions

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