N3 - Inverse operations and order of operations

Operations that undo each other

Two friends work out $18 - 6 \div 3 \times 2$. One gets $8$ and the other gets $14$. Can both answers be right? The expression needs a shared reading rule, just as a sentence needs punctuation. We will settle it once that rule is in place.

If you add $8$, subtracting $8$ takes you back. If you multiply by $5$, dividing by $5$ takes you back. These are inverse operations. To undo a chain, start with its last operation, because that is the one acting on the final result.

[DIAGRAM: asset_name: Undoing a two-step number machine; asset_slug: aqa8300_n19_inverse_machine; caption: Multiply by 4 then add 7 goes from 9 to 43; undoing subtracts 7 then divides by 4; description: A top row of three boxes shows 9, 36 and 43 joined left to right by multiply by 4 and add 7. A lower row shows 9, 36 and 43 joined right to left by divide by 4 and subtract 7. Blue arrows show the forward route and sage arrows the reverse route.]

The reverse route must use the reverse order. Starting at $43$, dividing by $4$ first would undo a multiplication that happened before the later addition; it would not return $9$.

An operation can be checked by asking whether its inverse restores the input. These inverse relationships are not on the formulae sheet: learn them.

The next example changes only the length of the process: an addition comes after the multiplication. Undo that last step first.

There is a boundary case worth noticing. Multiplying either $3$ or $100$ by zero gives zero. The output no longer tells you which input was used.

In the next chain, change the final amount while keeping the same reverse-order idea.

Reciprocals and cancellation

Some operations can cancel before you calculate. In $27 \times 8 \div 8$, multiplying and then dividing by $8$ undo each other, leaving $27$.

A reciprocal is a number that makes a product of $1$. The reciprocal of $5$ is $\frac{1}{5}$, because $5 \times \frac{1}{5} = 1$. The reciprocal of $\frac{1}{4}$ is $4$; the reciprocal of $-2$ is $-\frac{1}{2}$. Zero has no reciprocal, because zero times any number is zero, never $1$.

Cancellation means replacing a factor and its inverse by $1$. It does not mean deleting any matching numbers you happen to see.

The next example changes only the final divisor. It cancels part of the multiplier rather than all of it.

Addition has its own inverse cancellation: $64 + 19 - 19 = 64$. But in $(64+19) \div 19$, the divisor acts on the whole sum. The $+19$ and $\div19$ are not inverses, so they do not cancel.

The multiplier now has a different cancelling factor. Show the smaller calculation you use.

Which operation comes first?

An expression has one conventional reading. In $5 + 3 \times 4$, the multiplication belongs together: three groups of $4$, with $5$ added afterwards. That gives $17$, not $32$.

The order of operations is the priority rule for reading a calculation. Brackets come first; powers and roots come next; multiplication and division share a priority; addition and subtraction share the final priority. Work from left to right when operations have the same priority.

You may know the reminder BIDMAS or BODMAS. Neither means that division always comes before multiplication, or that addition always comes before subtraction. The equal-priority pairs matter. These conventions are not on the formulae sheet: learn them.

The next example changes only one feature: the product includes a division as well. Resolve that equal-priority chain from left to right.

The same left-to-right rule applies to addition and subtraction: $10-3+2=7+2=9$, not $10-5=5$. A subtraction sign stays with the quantity being subtracted.

We can now resolve the opening disagreement. $18-6\div3\times2=18-2\times2=18-4=14$. The friend who got $8$ worked through every sign from left to right, including subtraction too early.

Brackets, powers, roots and grouped expressions

Brackets tell you to treat their contents as one quantity. Powers, root signs and fraction bars also tell you exactly which quantity an operation acts on.

In $(3+2)^2$, first make $5$, then square it: $25$. In $3+2^2$, square only the $2$, giving $3+4=7$. A square means multiply the base by itself; a square root reverses squaring for a non-negative input. For example, $\sqrt{36}=6$, because $6^2=36$. Here we use familiar squares; the full set of powers and roots comes later.

A root bar groups everything underneath it: $\sqrt{9+16}=\sqrt{25}=5$, whereas $\sqrt9+16=3+16=19$. A fraction bar groups its complete numerator and denominator: $\frac{8+4}{6-2}=(8+4)\div(6-2)=3$. The reciprocal of $(3+2)$ is $\frac{1}{3+2}=\frac15$, not $\frac13+2$. These notation rules are not on the formulae sheet: learn them.

The next example changes only the squared quantity: a new pair of brackets makes the whole difference the base of the power.

A minus sign needs the same care: $(-4)^2=16$, because both factors are $-4$, but $-4^2=-16$, because the minus is outside the square. Also, $\sqrt{16}=4$, not $\pm4$: the square-root symbol means the non-negative square root. Solving an equation with a square is a later, different task.

Here is an unusual coincidence: $1+2+3=6$ and $1\times2\times3=6$. The same trick fails when you include $4$: $1+2+3+4=10$, while $1\times2\times3\times4=24$. Agreeing in one example does not make two operations interchangeable.

Read the brackets before calculating. The next check changes the numbers from Worked example 8.

Quick Recap

• Inverse operations undo each other; undo a chain in reverse order.
• The reciprocal of a non-zero number $a$ is $\frac1a$. Cancel inverse operations only when they act on the same quantity.
• Brackets come first, then powers and roots, then multiplication and division, then addition and subtraction.
• Equal-priority operations go from left to right. Division does not outrank multiplication.
• Brackets, fraction bars and root bars define complete groups; an exponent applies only to its stated base.

The opening calculation has the single conventional value $14$: complete $6\div3\times2$ before subtracting it from $18$.