Order of operations with powers and roots

Big math expressions look scary, but you can tame them by doing one step at a time in the right order.

Step 1: Solve everything inside brackets first. If brackets are nested inside each other, start with the innermost one and work your way out.

Step 2: Next, work out any powers and roots, like squares, cubes, and square roots.

Step 3: Then do all the multiplying and dividing, working from left to right.

Step 4: Finish with adding and subtracting, working from left to right.

Try it on 12 - 8 ÷ 4 + [(6 + 2) - 3] × 3. First solve the innermost bracket: 6 + 2 = 8. That leaves 12 - 8 ÷ 4 + [8 - 3] × 3. Solve the next bracket: 8 - 3 = 5, giving 12 - 8 ÷ 4 + 5 × 3. Now divide and multiply: 8 ÷ 4 = 2 and 5 × 3 = 15, giving 12 - 2 + 15. Finally add and subtract left to right: 12 - 2 = 10, then 10 + 15 = 25.

When a math expression has lots of operations mixed together, we need to agree on which one to do first. Otherwise, two people could solve the exact same expression and get two different answers.

So mathematicians made a rule everyone follows. Multiplication is really just repeated addition, so we do multiplication before addition. Powers are repeated multiplication, so we do powers before multiplication.

Grouping symbols like parentheses, brackets, and braces let you break this rule on purpose. Whatever is inside them gets solved first, before anything else.

Grouping symbols show you which part of an expression to solve first. Parentheses look like ( ), brackets look like [ ], and braces look like { }.

Sometimes these symbols are nested inside each other, like layers of a nesting doll. When that happens, always start with the symbol on the very inside and work your way out.

Try {3 + [2 × (4 + 1)]}. Start with the innermost parentheses: 4 + 1 = 5. Next, work on the brackets: 2 × 5 = 10. Finally, finish the braces: 3 + 10 = 13.

Calculators are built to follow the order of operations automatically, but you need to know the rules yourself so you can check your work and solve problems by hand.

The order goes like this: solve anything inside grouping symbols first, then handle powers and roots, then multiply and divide from left to right, and finish by adding and subtracting from left to right.

If an expression has grouping symbols nested inside other grouping symbols, always start with the innermost one.

Practicing this order on different expressions helps it become automatic, so you will not need to stop and think about which step comes next.

Every real number can be described as natural, whole, an integer, rational, or irrational, but no matter what kind of number you are working with, the same order of operations applies when you calculate.

Work out anything in parentheses first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right.

Real numbers also follow some useful properties. The inverse property says that every number has an opposite that combines with it to make zero or one. For multiplication, that opposite is called the reciprocal. For example, 4 and 1/4 are reciprocals of each other, because 4 × 1/4 = 1.