Order of Operations II Worksheets

Order of operations with exponents, division, brackets and negative bases. Grades 7–9.

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What these worksheets practise

Order of Operations II extends Order of Operations I with the rest of the rules: exponents come right after grouping symbols, multiplication and division share a rank and go left to right, and square brackets [ ][\ ] group an expression that already contains parentheses. Work from the innermost group outward.

The expressions are longer than in part I, and every worksheet cell has room for step-by-step working. Every division in a problem comes out even, so the answers are whole numbers.

  • Easy — one square or cube combined with multiplication, addition or subtraction.
  • Medium — two exponents or an exponent with brackets; some problems divide by a parenthesized expression.
  • Hard — brackets with parentheses inside, several exponents, division, and sometimes a negative base such as (−3)2(-3)^{2}.

Worked examples

Easy: 5+23×35 + 2^{3} \times 3

Exponent first, then multiply, then add:

5+23×3=5+8×3=5+24=295 + 2^{3} \times 3 = 5 + 8 \times 3 = 5 + 24 = 29

Medium: 6×4−[33÷(6+4−1)]6 \times 4 - [3^{3} \div (6 + 4 - 1)]

Start with the innermost parentheses, then the exponent inside the brackets:

6×4−[27÷9]=6×4−3=24−3=216 \times 4 - [27 \div 9] = 6 \times 4 - 3 = 24 - 3 = 21

Hard: [32×(10−4)]÷2+(−3)2[3^{2} \times (10 - 4)] \div 2 + (-3)^{2}

[9×6]÷2+9=54÷2+9=27+9=36[9 \times 6] \div 2 + 9 = 54 \div 2 + 9 = 27 + 9 = 36

Note that (−3)2=(−3)(−3)=9(-3)^{2} = (-3)(-3) = 9.

Check your answers

Write one step per line and make sure each line is shorter than the one before. If a line gets longer, something was expanded instead of evaluated.

Common mistakes

  • Multiplying the base by the exponent. 23=2×2×2=82^{3} = 2 \times 2 \times 2 = 8, not 6.
  • Multiplying before the exponent. 2×32=2×9=182 \times 3^{2} = 2 \times 9 = 18, not 62=366^{2} = 36.
  • Losing the sign of a negative base. (−3)2=9(-3)^{2} = 9, but −32=−9-3^{2} = -9: without parentheses, only the 3 is squared.
  • Doing multiplication before division. They have equal rank: 24÷4×2=6×2=1224 \div 4 \times 2 = 6 \times 2 = 12, not 24÷8=324 \div 8 = 3.

Sample problems

These problems come from the same generator as the worksheet above, three at each difficulty level. Press Generate for a fresh set.

Easy

  1. 4×(3+42)−3=4 \times (3 + 4^{2}) - 3 =
  2. 52−2+4×2=5^{2} - 2 + 4 \times 2 =
  3. (3+3)2−2×2=(3 + 3)^{2} - 2 \times 2 =

Medium

  1. 5×2−[52÷(5+4−4)]=5 \times 2 - [5^{2} \div (5 + 4 - 4)] =
  2. (32−6)×2+33=(3^{2} - 6) \times 2 + 3^{3} =
  3. (22−7)×2+23=(2^{2} - 7) \times 2 + 2^{3} =

Hard

  1. [32×(11−3)]÷4+(−6)2=[3^{2} \times (11 - 3)] \div 4 + (-6)^{2} =
  2. 5+22−[6+4×(33÷3)]=5 + 2^{2} - [6 + 4 \times (3^{3} \div 3)] =
  3. [32×(7−2)]÷3+(−4)2=[3^{2} \times (7 - 2)] \div 3 + (-4)^{2} =
Show answers
  1. 73
  2. 31
  3. 32
  4. 5
  5. 33
  6. 2
  7. 54
  8. -33
  9. 31

Tips for teachers and parents

  • Have students circle the innermost group before starting a Medium or Hard problem.
  • A short list of squares up to 626^{2} and cubes up to 434^{3} keeps attention on the order rather than on arithmetic.
  • The difference between (−3)2(-3)^{2} and −32-3^{2} is worth a separate class discussion; it comes back in algebra and on calculators.

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