The correct order of operations is grouping symbols, exponents, multiplication and division, then addition and subtraction.

Title: The Right Way to Tackle Numbers: Why Grouping Symbols Come First

Let me ask you something: when you see a big math expression, do you ever feel like you’ve walked into a kitchen with a recipe you’ve never seen before? If you’re cooking up a solution on an HSPT math task or just trying to make sense of numbers for yourself, the trick is to follow a clear order. Not a guesswork game, but a dependable sequence that keeps every calculation honest. The order you’ll want to memorize is simple, almost like a recipe: grouping symbols first, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). It’s the standard sequence that removes guesswork and keeps every step aligned with the math itself.

Let’s unpack what that order really means and why it matters. We’ll mix a little intuition with a couple of concrete examples, and I’ll throw in a few practical tips that you can use without turning math into a lab experiment.

Grouping Symbols First: Tidy the Interior, Then Get Serious

Think of grouping symbols as the door to the problem. Parentheses, brackets, and other kinds of grouping marks tell you which parts to handle first. If you skip this step, you’re like a chef who starts by chopping onions after you’ve already put the pot on the stove with a half-cooked sauce. The interior of grouping symbols gets treated as a mini-problem that must be solved before you touch anything outside.

Here’s a straightforward demonstration. Consider the expression:

(3 + 2)^2 × 2 + 5

• Start inside the grouping symbols: 3 + 2 = 5

• Replace that chunk: (3 + 2)^2 becomes 5^2

• Now you’re at 25 × 2 + 5

This step is all about giving you a clean slate. Grouping symbols ensure that the parts of the expression that could change everything about the rest are settled before you move on. It’s the same principle you’d use when following a family recipe: you measure out the exact ingredients inside the parentheses before you even think about simmering, stirring, or serving.

Exponents: Powers, Then Power Plays

Once you’ve cleared grouping symbols, exponents come into play. Exponents aren’t just fancy notation; they’re compact shortcuts that can drastically shift the scale of your numbers. Treat them with respect, because they set the pace for what follows.

Using the same example, after you resolve the grouping symbols you arrive at:

5^2 × 2 + 5

Now you apply the exponent rule: 5^2 = 25. The expression becomes:

25 × 2 + 5

If you skip this step or treat the exponent like just another multiplication, you’ll end up with the wrong total. Exponents sit neatly in their own lane, and that lane is before multiplication and division.

Multiplication and Division: Left to Right, Then Left to Right Again

Multiplication and division sit at the same level of priority. The rule “from left to right” is not a nitpick—it keeps the calculation consistent, especially when both operations appear in the same expression. If you see 25 × 2 ÷ 5, you do the multiplication first (25 × 2 = 50), then the division (50 ÷ 5 = 10). The same goes for additions and subtractions that follow later.

Back to our example:

25 × 2 + 5

Here you multiply first: 25 × 2 = 50, and then you add:

50 + 5 = 55

That final result, 55, comes directly from honoring the multiplication-before-addition rule. If you swapped the order and did addition first, you’d get a completely different number, which proves why the rules exist in the first place.

Addition and Subtraction: The Final Step, Left to Right

Addition and subtraction share the same priority level and follow the left-to-right rule. By the time you reach this stage, you’re essentially stringing together solved pieces into the final total. It’s a calm finish line after a careful build-up.

Continuing with the earlier steps, you end up with:

50 + 5 = 55

That’s the clean, correct conclusion when you stick to the four-part sequence. The certainty you gain from this order can be surprisingly comforting, especially when you’re sorting through more elaborate expressions on a test.

Why the Order Keeps You Honest

You might wonder why we bother with grouping symbols and exponents at all. Why not jump straight to the multiplying and adding, just to see what comes out? The short answer: math is a language with its own grammar. If you ignore grouping symbols or shove exponents to the back, you’re breaking the rules of that language. The result becomes ambiguous, or worse—wrong.

A quick mental test helps here. Take a tricky expression like:

(2 + 3)^2 − 4 ÷ 2

If you skip the grouping step and try to do things from left to right willy-nilly, you’ll end up with a different number than if you first resolve inside the parentheses and then handle the exponent. The order gives you a single, unambiguous result every time. That predictability is a huge advantage in the kind of math you’ll encounter on the HSPT math section and beyond.

Real-world anchors: why this matters beyond the page

If you’ve ever built something with instructions, you know the power of following a sequence. A board game with shuffled steps feels chaotic until someone announces, “First, read the rules; then set up the board.” Math is similar. The order of operations is a guideline that helps you reason clearly, communicate your process, and check your work more easily. When you see a long expression, you won’t be paralyzed by fear or swayed by a guess. You’ll have a reliable plan you can follow step by step.

A few quick tips to keep you sharp

• Visualize layers: Treat grouping symbols as the innermost layers of an onion. Peel them first, then move outward to exponents, then multiplications/divisions, and finally additions/subtractions.

• Write it down: For anything a bit longer, jot the steps on paper. It’s common to mistype or misread fast in your head. A clear scratch work line helps you stay honest.

• Check your result by reversing: If you’re not sure, try to work backward from the end. Do you get back to the original expression if you reverse operations? If not, you likely slipped somewhere.

• Use a mnemonic wisely: PEMDAS or BODMAS can help you remember, but don’t treat them as rigid rituals—you still need to apply the steps in the proper order.

A couple of extra mini-examples to reinforce the pattern

• Example A: 7 + 2 × 3^2

• Exponent first: 3^2 = 9

• Multiplication next: 2 × 9 = 18

• Then addition: 7 + 18 = 25

• Final: 25

This shows how a quick exponent can change the picture dramatically before any adding happens.

• Example B: (6 − 2)^3 ÷ 2

• Grouping first: (6 − 2) = 4

• Exponent next: 4^3 = 64

• Division last: 64 ÷ 2 = 32

• Final: 32

Grouping changed the base entirely, meaning the later steps work on a different number.

A small note on flexibility and intuition

You’ll hear people describe math as rigid, but there’s room for intuition—especially when expressions look messy or involve mixed operations. The key is to keep the four-step rhythm in your head. Once you’re comfortable with it, you’ll notice patterns—like how exponents often flip the scale quickly, or how grouping signals can protect you from a cascading misstep.

Bringing it together: the core takeaway

When you’re solving a numerical expression, the order to rely on is straightforward and powerful:

Grouping Symbols, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)

That sequence isn’t just a rule to memorize; it’s a practical tool that makes mathematical thinking transparent and reliable. It’s the same logic you’d apply to any math you encounter, be it a homework problem, a timed test question, or something you’re exploring for its own sake. The heart of it is simple: resolve the tightest, most influential parts first, then move outward with the operations that build the final answer.

A few closing thoughts

If you’re reading this and thinking, “this makes sense, but I still feel a little unsure,” that’s totally normal. Like any skill, fluency comes with practice and steady feedback. The more expressions you walk through in this order, the quicker your eyes will recognize what comes next. And yes, those quick checks—like testing a simplified version or reversing the steps—can be surprisingly reassuring.

Takeaway checklist for easy reference

• Start with anything inside grouping symbols.

• Apply exponents next.

• Do all multiplication and division from left to right.

• Finish with addition and subtraction from left to right.

Next time you’re faced with a long expression, imagine you’re stepping through a clean, well-lit hallway. The walls are the grouping symbols; the stairs are exponents; the doors along the way are multiplication and division; and the final corridor leads you to the destination, the clear answer. The path is simple when you follow the sequence, and suddenly the math stops feeling like a maze and starts feeling like a map.

If you’re curious to see more examples like this, you’ll find that each problem becomes a little puzzle with a predictable rhythm. And that rhythm? It’s a reliable compass for the math you’ll encounter in the HSPT math section, helping you reason clearly and stay confident as you work through each item.