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Difficulty: MediumOrder of Operations and Number Properties

Evaluate the four mathematical expressions below, and arrange them in order from least to greatest value.

  1. 1−24+3×(5−8)-2^4 + 3 \times (5 - 8)
  2. 2−∣−6−22×3∣+10-| -6 - 2^2 \times 3 | + 10
  3. 3(3−5)3÷4−2(3 - 5)^3 \div 4 - 2
  4. 42×(−3)2−42÷82 \times (-3)^2 - 4^2 \div 8

Answer

The correct order of expressions from least to greatest is the expression evaluating to −25-25, followed by the expression evaluating to −8-8, then the expression evaluating to −4-4, and finally the expression evaluating to 1616.
Evaluating each expression using the correct order of operations yields −25-25 for the expression containing the negative base exponent, −8-8 for the expression containing the absolute value, −4-4 for the expression containing the cubed term, and 1616 for the expression containing the squared term. Ordering these values from least to greatest results in the sequence −25<−8<−4<16-25 < -8 < -4 < 16.

Step-by-Step Solution

1
Evaluate the first expression: 2×(−3)2−42÷82 \times (-3)^2 - 4^2 \div 8.
Value is 1616.
Exponents are calculated first: (−3)2=9(-3)^2 = 9 and 42=164^2 = 16. Then, perform multiplication and division: 2×9=182 \times 9 = 18 and 16÷8=216 \div 8 = 2. Finally, subtract: 18−2=1618 - 2 = 16.
2
Evaluate the second expression: −24+3×(5−8)-2^4 + 3 \times (5 - 8).
Value is −25-25.
Parentheses are evaluated first: 5−8=−35 - 8 = -3. Next, evaluate the exponent: −24=−16-2^4 = -16 (since the exponent applies to the base 22 before negation). Perform multiplication: 3×(−3)=−93 \times (-3) = -9. Finally, add: −16+(−9)=−25-16 + (-9) = -25.
3
Evaluate the third expression: (3−5)3÷4−2(3 - 5)^3 \div 4 - 2.
Value is −4-4.
Parentheses are evaluated first: 3−5=−23 - 5 = -2. Next, cube the result: (−2)3=−8(-2)^3 = -8. Perform division: −8÷4=−2-8 \div 4 = -2. Finally, subtract: −2−2=−4-2 - 2 = -4.
4
Evaluate the fourth expression: −∣−6−22×3∣+10-| -6 - 2^2 \times 3 | + 10.
Value is −8-8.
Within the absolute value bars, evaluate the exponent first: 22=42^2 = 4, then multiply: 4×3=124 \times 3 = 12. Subtract to find the absolute value interior: −6−12=−18-6 - 12 = -18. The absolute value of −18-18 is 1818, which is negated to −18-18. Finally, add 1010 to get −18+10=−8-18 + 10 = -8.
5
Order the final values from least to greatest.
−25<−8<−4<16-25 < -8 < -4 < 16.
Comparing the values −25-25, −8-8, −4-4, and 1616 on the number line determines their relative size.

Key Concept

Order of operations rules dictate that expressions are evaluated in the sequence of Parentheses/Grouping symbols, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Unary negation is performed after exponentiation unless grouped by parentheses.
Estimated Time:1m 30s
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Topics

Order of Operations and Number Properties

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