doing math

courses › Pre-Algebra

Order of operations

level 11 course

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

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

Learn first (about 3 minutes)

Opens at level 11.

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Builds on: Negative numbers (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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.

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.

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

Example: 20 − 12 ÷ 4 + 2 × 3

  1. × and ÷ first: 12 ÷ 4 = 3, and 2 × 3 = 6.
  2. 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.

  1. Work out the bracket to one number.
  2. Apply the power to that number.
  3. Then multiply, and add or subtract last.
worked example

Example: 3 × (1 + 4)² − 5

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

Answer: 70

Where the minus sign sits

A minus sign right next to a power.

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

Example: −4² + (−2)³

  1. −4² is the negative of 4 × 4, so −16.
  2. (−2)³ = (−2) × (−2) × (−2) = −8.
  3. −16 + (−8) = −24.

Answer: −24

Tips by skill

  • Tip× before +: Do every × and ÷ first, left to right. Then do the + and − from left to right.
  • TipBrackets & powers: Bracket to one number first, then the power, then ×, then + or − last.
  • TipSigns & powers: Is the negative inside a bracket? Then the power includes the sign. No bracket? Power first, then make it negative.

Watch out for

  • 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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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