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Difficulty: HardOrder of Operations and Number Properties

An expression is shown below.

23×22−4×(1−3)3∣2−23∣−10 \frac{2^3 \times 2^2 - 4 \times (1 - 3)^3}{|2 - 2^3| - 10}

What is the value of this expression when it is simplified using the standard order of operations?

  1. A
    -24
  2. -16Answer
  3. C
    -4
  4. D
    0
  5. E
    56

Answer

The correct answer is −16-16.
To find the value of the expression, follow the standard order of operations (PEMDAS). First, simplify terms with exponents and parentheses: 23=82^3 = 8, 22=42^2 = 4, and (1−3)3=(−2)3=−8(1 - 3)^3 = (-2)^3 = -8. The numerator becomes 8×4−4×(−8)=32−(−32)=648 \times 4 - 4 \times (-8) = 32 - (-32) = 64. In the denominator, evaluate the exponent first to get 2−8=−62 - 8 = -6, then apply the absolute value to get ∣−6∣=6|-6| = 6, and finally subtract 1010 to get 6−10=−46 - 10 = -4. Dividing the numerator by the denominator gives 64−4=−16\frac{64}{-4} = -16.

Step-by-Step Solution

1
Evaluate the exponents in the numerator and the denominator, including the term inside the parenthesis.
In the numerator, 23=82^3 = 8, 22=42^2 = 4, and (1−3)3=(−2)3=−8(1 - 3)^3 = (-2)^3 = -8. In the denominator, 23=82^3 = 8.
According to the order of operations (PEMDAS), operations inside grouping symbols (parentheses) and exponentiation must be performed before multiplication, division, addition, or subtraction.
2
Perform the multiplications in the numerator and evaluate the expression inside the absolute value in the denominator.
The numerator becomes 8×4−4×(−8)=32−(−32)8 \times 4 - 4 \times (-8) = 32 - (-32). The denominator becomes ∣2−8∣−10=∣−6∣−10|2 - 8| - 10 = |-6| - 10.
Multiplication has priority over addition and subtraction. In the denominator, the expression inside the absolute value acts as a grouping symbol and is simplified first.
3
Perform addition and subtraction in both the numerator and the denominator, including applying the absolute value.
The numerator is 32+32=6432 + 32 = 64. The denominator is 6−10=−46 - 10 = -4.
Subtracting a negative number is equivalent to adding its positive counterpart: 32−(−32)=32+3232 - (-32) = 32 + 32. The absolute value of −6-6 is 66, so the denominator simplifies to 6−10=−46 - 10 = -4.
4
Divide the simplified numerator by the simplified denominator.
64−4=−16\frac{64}{-4} = -16.
The fraction bar represents division, which is the final operation to perform to find the value of the expression.

Key Concept

Order of Operations and Number Properties

Alternative Method

Instead of evaluating 23×222^3 \times 2^2 separately, the laws of exponents can be used to simplify the term first: 23×22=23+2=25=322^3 \times 2^2 = 2^{3+2} = 2^5 = 32.
Estimated Time:1m 30s
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Topics

Order of Operations and Number Properties

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