Powers in the Order of Operations

The index belongs to one number

In $3\times 4^2$3 × 4², the small 2 is the index, and it belongs only to the number it sits on, the 4. So $4^2$4² means 4 × 4 = 16, and the 3 is not squared.

Work out the power first, then multiply: $3\times 4^2=3\times 16=48$3 × 4² = 3 × 16 = 48. A power is a short way of writing a multiplication, 4 × 4, so it is worked out as one number before anything else touches it.

$4^2$4² is one 4 by 4 square of 16 tiles, colored. $3\times 4^2$3 × 4² is three of those squares: 3 × 16 = 48 tiles.

Brackets still come first

Brackets change what the index belongs to. In ${\left(3\times 4\right)}^2$(3 × 4)², the index sits on the bracket, so the whole of 3 × 4 is squared. Work out the bracket first: 3 × 4 = 12. Then square it: $12^2=12\times 12=144$12² = 12 × 12 = 144.

So $3\times 4^2=48$3 × 4² = 48 but ${\left(3\times 4\right)}^2=144$(3 × 4)² = 144. The numbers and signs are the same; the brackets alone make the difference.

${\left(3\times 4\right)}^2$(3 × 4)² is a 12 by 12 square of 144 tiles. The colored part is 4 rows of 12, the 48 tiles of $3\times 4^2$3 × 4², only a third of it.

Before adding and subtracting too

A power also comes before adding and subtracting. In $5+2^3$5 + 2³, work out the power first: $2^3=2\times 2\times 2=8$2³ = 2 × 2 × 2 = 8. Then add: 5 + 8 = 13.

Adding first would give ${\left(5+2\right)}^3=7^3=343$(5 + 2)³ = 7³ = 343, which is a different calculation. And the index is not a multiplier: $2^3$2³ is 8, not 2 × 3 = 6.

Roots at the same step

A root undoes a power, and it is worked out at the same step as a power. In $20-\sqrt{36}\times 2$20 − √36 × 2, take the root first: $\sqrt{36}=6$√36 = 6, because 6 × 6 = 36. The line becomes 20 − 6 × 2.

Then multiply: 6 × 2 = 12, and the line becomes 20 − 12. Last, subtract: 20 − 12 = 8.

The root and the multiplication together make 6 × 2 = 12, so the subtraction takes 12 away from 20 and lands on 8.

The order of operations, with powers

Powers and roots add one step to the order of operations. First, work out anything in brackets. Next, work out powers and roots. Then do every × and ÷, from left to right. Last, do every + and −, from left to right.

Here is every step in one line: $3^2+4\times \left(7-2\right)$3² + 4 × (7 − 2). The bracket comes first: 7 − 2 = 5, so the line becomes $3^2+4\times 5$3² + 4 × 5. The power comes next: $3^2=9$3² = 9, so the line becomes 9 + 4 × 5. Then multiply: 4 × 5 = 20. Last, add: 9 + 20 = 29.

In a formula

Formulas in science often hold a power, and the same order applies after you substitute the numbers. In the next problem the speed is squared, so work out the square before you multiply by anything else.