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Order of operations

lesson · about 3 minutes

Brackets, powers, then × and ÷, then + and −.

Best in your head · no pen, no calculator why?

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the idea

An expression like 3 + 4 × 5 needs one agreed reading, or two people get two answers. The agreement: brackets first, then powers, then × and ÷, then + and −. Between × and ÷, or between + and −, go left to right. So 3 + 4 × 5 is 3 + 20 = 23, not 35.

more of the idea

One trap needs its own rule. In −3², the power belongs to the 3 alone, so it means the negative of 3 × 3, which is −9. To square the negative number, the bracket has to be written: (−3)² is 9.

techniques

Work in the agreed order Any expression with more than one operation.

Brackets first: work out everything inside them.
Then powers: 5² means 5 × 5, not 5 × 2.
Then × and ÷, from left to right.
Last, + and −, from left to right.

example 20 − 12 ÷ 4 + 2 × 3

× and ÷ first: 12 ÷ 4 = 3, and 2 × 3 = 6.
Then left to right: 20 − 3 + 6 = 23.

Answer: 23

Bracket, then its power A bracket with a power on it, like 2 × (3 + 5)².

Work out the bracket to one number.
Apply the power to that number.
Then multiply, and add or subtract last.

example 3 × (1 + 4)² − 5

Bracket: 1 + 4 = 5.
Power: 5 × 5 = 25.
Multiply: 3 × 25 = 75.
Subtract: 75 − 5 = 70.

Answer: 70

Where the minus sign sits A minus sign right next to a power.

No bracket, as in −4²: square the 4, then make it negative.
A bracket, as in (−4)²: multiply the negative number by itself.
An even power of a negative number is positive; an odd power stays negative.

example −4² + (−2)³

−4² is the negative of 4 × 4, so −16.
(−2)³ = (−2) × (−2) × (−2) = −8.
−16 + (−8) = −24.

Answer: −24

watch out for · 4 traps

Working straight left to right. In 3 + 4 × 5 the multiplication comes first, so the answer is 23, not 35.
Multiplying before squaring. In 2 × (3 + 5)², square the 8 first, then double it.
Reading −3² as (−3)². Without a bracket the square belongs to the 3, so the answer is negative.
Treating a power as times the exponent: 6² is 36, not 12.

practice

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× before +

worked example

9 + 11 × 7

× and ÷ come before + and −: 11 × 7 = 77.
Then 9 + 77 = 86.

Answer: 86

Brackets & powers

worked example

(4 + 4) × 8 − 6²

Brackets first: 4 + 4 = 8.
Power: 6² = 36.
Multiply: 8 × 8 = 64.
Then 64 − 36 = 28.

Answer: 28

Signs & powers

worked example

−4² − (−2)²

−4² means −(4²) = −16.
(−2)² = (−2) × (−2) = 4.
Total: −16 − 4 = −20.

Answer: -20