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Difficulty: Very hardEvaluating Algebraic Expressions

If m=−3m = -3, n=18n = \frac{1}{8}, and p=−2p = -2, what is the value of the algebraic expression m−2−n−1/3⋅p−3m^{-2} - n^{-1/3} \cdot p^{-3}? Express your answer as a simplified fraction.

Answer:The value of the expression is 【13/36】.

Answer

The correct answer is 13/36.
Evaluating each term individually gives m−2=19m^{-2} = \frac{1}{9}, n−1/3=2n^{-1/3} = 2, and p−3=−18p^{-3} = -\frac{1}{8}. Substituting these values into the expression yields 19−2⋅(−18)\frac{1}{9} - 2 \cdot \left(-\frac{1}{8}\right). Following the order of operations, we first perform the multiplication: 2⋅(−18)=−142 \cdot \left(-\frac{1}{8}\right) = -\frac{1}{4}. We then subtract this result from the first term: 19−(−14)=19+14=1336\frac{1}{9} - \left(-\frac{1}{4}\right) = \frac{1}{9} + \frac{1}{4} = \frac{13}{36}.

Step-by-Step Solution

1
Evaluate m−2m^{-2} when m=−3m = -3.
m−2=(−3)−2=1(−3)2=19m^{-2} = (-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}
A negative exponent represents the reciprocal of the base raised to the positive power, and a negative base raised to an even power yields a positive result.
2
Evaluate n−1/3n^{-1/3} when n=18n = \frac{1}{8}.
n−1/3=(18)−1/3=(8−1)−1/3=81/3=2n^{-1/3} = \left(\frac{1}{8}\right)^{-1/3} = \left(8^{-1}\right)^{-1/3} = 8^{1/3} = 2
Apply the negative exponent rule to find the reciprocal of the fraction, then find the cube root of 8.
3
Evaluate p−3p^{-3} when p=−2p = -2.
p−3=(−2)−3=1(−2)3=−18p^{-3} = (-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8}
A negative exponent represents the reciprocal, and a negative base raised to an odd power yields a negative result.
4
Substitute the evaluated terms back into the original algebraic expression.
19−2⋅(−18)\frac{1}{9} - 2 \cdot \left(-\frac{1}{8}\right)
Replace each variable expression with its calculated numerical value.
5
Perform the multiplication before subtraction following the order of operations.
2⋅(−18)=−28=−142 \cdot \left(-\frac{1}{8}\right) = -\frac{2}{8} = -\frac{1}{4}
The order of operations (PEMDAS/GEMS) dictates that multiplication must be performed before subtraction.
6
Subtract the product from the first term.
19−(−14)=19+14=436+936=1336\frac{1}{9} - \left(-\frac{1}{4}\right) = \frac{1}{9} + \frac{1}{4} = \frac{4}{36} + \frac{9}{36} = \frac{13}{36}
Subtracting a negative value is equivalent to addition. Find a common denominator to add the fractions.

Key Concept

Evaluating algebraic expressions involving negative bases, negative exponents, fractional exponents, and the order of operations.
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Topics

Evaluating Algebraic Expressions

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