1.03a - Order of operations | OCR GCSE Mathematics

Why does $6+2\times5$ mean $16$ , rather than $40$ ? Learn the agreed order for brackets, powers, roots and the four operations, so that a calculation has one clear meaning.

Powers before products

If six tickets cost £1 each and two tickets cost £5 each, what is the total cost? The calculation is $6+2\times5=16$ : the two £5 tickets form a group before their cost is added to £6. Working strictly from left to right would instead price eight £5 tickets.

An expression is a written calculation, such as $6+2\times5$ . The agreed order of operations tells you which pieces to calculate first. Powers and roots describe a single number within that expression; evaluate them before multiplying or adding other terms.

Key idea

Work inside brackets first. Then evaluate powers, roots and reciprocals. Then do multiplication and division. Finally do addition and subtraction. Multiplication and division share a priority; addition and subtraction share a priority.

For example, $\sqrt{16}=4$ , and $3^{-1}=\frac13$ , the reciprocal of $3$ . A negative index asks for a reciprocal; it does not make the answer negative. These values must be found before the surrounding products. Division by zero remains undefined whatever the order of operations.

Worked example 1: a power within a product

Question. Work out $12\times3^2+\sqrt{16}$ .

Plan. Evaluate the power and root, then multiply, then add.

\begin{aligned} 12\times3^2+\sqrt{16} &= 12\times9+4 && \text{power and root first} \\ &=108+4 && \text{multiply} \\ &=112 && \text{add last} \end{aligned}

Answer. $112$ .

Check. Nine groups of twelve make $108$ ; adding the root, $4$ , gives $112$ . Squaring $4$ also gives the original radicand, $16$ .

Now change just the index from $2$ to $-1$ . The order stays the same, but the powered number becomes a reciprocal.

Worked example 2: a reciprocal within a product

Question. Work out $12\times3^{-1}+\sqrt{16}$ .

Plan. Replace the negative power with its reciprocal before multiplying.

\begin{aligned} 12\times3^{-1}+\sqrt{16} &=12\times\frac13+4 && \text{reciprocal and root} \\ &=4+4 && \text{multiplying by }\frac13\text{ divides by 3} \\ &=8 && \text{add last} \end{aligned}

Answer. $8$ .

Check. Multiplying $4$ by $3$ restores $12$ , so the reciprocal product is $4$ , not $-36$ .

Keeping each stage on its own line makes the chosen order visible.

Watch out

$6+2\times5=8\times5$ is wrong: it performs the addition too early. The correct first step is $6+10$ .

Use the same priorities with new numbers.

Working inside brackets

Brackets turn several operations into one grouped quantity. In $(6+2)\times5$ , all eight tickets cost £5, so the total is £40. These brackets deliberately change the opening calculation.

Apply the same priority rules inside brackets. If there are brackets within brackets, start with the innermost group. A fraction bar also groups its whole numerator and whole denominator: $\frac{8+4}{2+1}=(8+4)\div(2+1)$ . A root sign groups everything beneath its bar: $\sqrt{9+16}=\sqrt{25}=5$ , whereas $\sqrt9+16=19$ .

Worked example 3: calculating within a group

Question. Work out $4\times(3+2^2)$ .

Plan. Finish the bracket, using the power first, then multiply by four.

\begin{aligned} 4\times(3+2^2)&=4\times(3+4) && \text{power inside bracket} \\ &=4\times7 && \text{finish bracket} \\ &=28 && \text{multiply outside} \end{aligned}

Answer. $28$ .

Check. Four groups each containing $3+4$ have $12+16=28$ altogether.

Move the closing bracket so that the square applies to the whole sum. Only the grouping changes; that changes the number being squared.

Worked example 4: squaring the whole bracket

Question. Work out $4\times(3+2)^2$ .

Plan. Add inside the bracket, square its value, then multiply.

\begin{aligned} 4\times(3+2)^2&=4\times5^2 && \text{bracket first} \\ &=4\times25 && \text{square the whole 5} \\ &=100 && \text{multiply last} \end{aligned}

Answer. $100$ .

Check. $(3+2)^2$ means $(3+2)\times(3+2)=5\times5=25$ , and four copies make $100$ .

You can also choose brackets to make an equality true. For instance, inserting one pair into $3\times8-2=18$ gives $3\times(8-2)=18$ . Check your proposed grouping by evaluating it; keep all the original numbers and operations.

Watch out

$(3+2)^2$ is $25$ , not $3+2^2=7$ . The position of the bracket decides what the index applies to.

Follow the grouping before looking at the outside multiplier.

Operations with equal priority

Multiplication does not automatically come before division. They occupy the same level, so work from left to right when neither is grouped by brackets. Addition and subtraction also share a level.

Thus $18-7+2$ means $(18-7)+2=13$ . Doing the addition first gives $18-(7+2)=9$ , which silently inserts different brackets. Memory aids for operation order can be useful, but their letter order must not separate operations that share a priority.

Worked example 5: division followed by multiplication

Question. Work out $24\div6\times2$ .

Plan. Work along the equal-priority operations from left to right.

\begin{aligned} 24\div6\times2&=4\times2 && \text{leftmost operation first} \\ &=8 && \text{then multiply} \end{aligned}

Answer. $8$ .

Check. $24\div6$ is $4$ ; doubling that quotient gives $8$ .

Now bracket the two numbers after the division sign. That single change makes the entire product the divisor.

Worked example 6: a grouped divisor

Question. Work out $24\div(6\times2)$ .

Plan. Finish the divisor in brackets before dividing.

\begin{aligned} 24\div(6\times2)&=24\div12 && \text{bracket has priority} \\ &=2 && \text{divide by the whole 12} \end{aligned}

Answer. $2$ .

Check. $2\times12=24$ , so $2$ is the quotient when the divisor is $12$ .

Writing a fraction bar would also make the distinction clear: $\frac{24}{6}\times2=8$ , while $\frac{24}{6\times2}=2$ .

Watch out

Do not treat $24\div6\times2$ as $24\div(6\times2)$ . The original calculation contains no instruction to multiply the divisor by two first.

The brackets in the next calculation are intentional.

Negative signs and powers

A negative base and a negative index mean different things. In $(-3)^2$ , the base is the whole negative number; in $3^{-2}$ , the base is positive and the negative index asks for a reciprocal.

Without brackets, $-3^2$ means the negative of $3^2$ . The power is evaluated before that leading minus sign: $-3^2=-(3\times3)=-9$ . By contrast, $(-3)^2=(-3)\times(-3)=9$ . With an odd positive index, $(-3)^3=-27$ , because three negative factors have a negative product.

Worked example 7: a minus outside the power

Question. Work out $-4^2+5$ .

Plan. Square four, apply the leading minus, then add five.

\begin{aligned} -4^2+5&=-(4\times4)+5 && \text{power before the minus} \\ &=-16+5 && \text{take the negative} \\ &=-11 && \text{add five} \end{aligned}

Answer. $-11$ .

Check. Subtracting $5$ from $-11$ gives $-16$ , the negative square required before the addition.

Add brackets around negative four. This changes only the base of the square.

Worked example 8: a negative base

Question. Work out $(-4)^2+5$ .

Plan. Multiply two copies of negative four, then add five.

\begin{aligned} (-4)^2+5&=(-4)\times(-4)+5 && \text{the whole base is repeated} \\ &=16+5 && \text{two negative factors} \\ &=21 && \text{add five} \end{aligned}

Answer. $21$ .

Check. Subtracting $5$ gives $16$ , which is the square of negative four.

This explains the opening puzzle: symbols describe a structure, rather than a list to process blindly. A pair of brackets can change the structure and the answer, even though every digit stays the same.

Watch out

$4^{-2}=\frac1{4^2}=\frac1{16}$ , not $-16$ . The sign in the index requests a reciprocal; a minus outside the power requests a negative value.

Identify the whole base before squaring.

Quick Recap

Finish groups before operations outside them; a fraction bar and a root bar also group terms.
Evaluate powers, roots and reciprocals before products and quotients.
Work left to right within multiplication/division, and within addition/subtraction.
Distinguish a negative base, a leading minus and a negative index.

The original total is $6+2\times5=16$ , because the two £5 tickets cost £10 before their cost is added to £6.

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1.03a - Order of operations

Powers before products

Working inside brackets

Operations with equal priority

Negative signs and powers

Quick Recap

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