How do you solve expressions with more than one type of bracket?

In one line

The rule

When solving expressions with multiple brackets (parentheses (), curly braces {}, and square brackets []), always start with the innermost bracket and work outward, following the order: ( ) → { } → [ ].

Let's understand it simply

Imagine you have three boxes, one inside the other. The smallest box is inside the medium box, which is inside the biggest box. To get to the smallest box, you must open the biggest box first, then the medium, and finally the small one. In maths, brackets work the same way. You solve the innermost bracket first, then the next one, and so on, until all brackets are gone.

How it works

1. Find the innermost bracket (the one inside all others).
2. Solve the operation inside this bracket.
3. Replace the bracket with its answer.
4. Move to the next bracket outside and repeat the process.
5. Continue until all brackets are removed and you have a simple expression.

Where you see it in real life

• Shopping bills: If you buy 2 sets of (3 pens + 2 pencils), you first add inside the bracket, then multiply.
• Recipe measurements: When doubling a recipe like 2 × (1 cup rice + ½ cup dal), you add inside first.
• Cricket scores: Calculating runs like 2 × [4 + (3 × 2)] means you do the multiplication inside the round bracket first.
• Class marks: If a teacher says, “Add 5 marks to (each student’s score + bonus),” you add inside the bracket first.

Why it matters

• Avoids mistakes: Doing brackets in the wrong order can give the wrong answer.
• Needed in algebra: Many equations and formulas use brackets.
• Used in science: Physics and chemistry problems often have nested brackets.
• Important for exams: Questions with brackets are common in board exams.

Worked example

Let’s solve: 2 × [3 + {4 × (2 + 1)}]

Given: 2 × [3 + {4 × (2 + 1)}]
Step 1: Solve innermost bracket (2 + 1) = 3
Step 2: Replace: 2 × [3 + {4 × 3}]
Step 3: Solve next bracket {4 × 3} = 12
Step 4: Replace: 2 × [3 + 12]
Step 5: Solve [3 + 12] = 15
Step 6: Replace: 2 × 15
Step 7: Final answer: 30

Common mistakes

• Solving brackets in the wrong order: Always start from the innermost bracket.
• Forgetting to multiply after removing brackets: Remember to multiply if there is a number outside the bracket.
• Missing signs: Don’t forget plus or minus signs inside or outside the brackets.

Remember it

Use the order: ( ) first, then { }, then [ ]. Think: "Smallest box first!"