Question bank — Order of operations — BODMAS - PEMDAS with nested brackets

In BODMAS, Division is always done before Multiplication. ::: False. The D and M are one single tier of equal rank; you resolve them from left to right, so whichever appears first goes first. In PEMDAS, Addition is always done before Subtraction. ::: False. Add and subtract share one tier; $8-3+2$ reads left to right as $5+2=7$, not $8-(3+2)=3$. A square-root sign  ​ counts as a grouping symbol, like a bracket. ::: True. Everything under the root is computed as one bundle first, exactly as if $\sqrt{9+16}=\sqrt{(9+16)}=\sqrt{25}=5$, not $3+4$. $-3^2$ and $(-3{)}^2$ are just two ways of writing the same number. ::: False. $-3^2=-(3^2)=-9$ because the power binds tighter than the minus; $(-3{)}^2=9$ because the bracket pulls the sign into the base first. See Negative numbers and the unary minus sign. Reordering a same-tier chain, like turning $20÷5\times 2$ into $20\times 2÷5$, never changes the answer. ::: False in general. $20÷5\times 2=8$ but $2÷5\times 20$ would need care; division is not commutative, so left-to-right reading is the only safe order — don't shuffle it. See Calculator vs mental arithmetic — parsing expressions. A fraction bar written vertically means you must add invisible brackets around the whole top and whole bottom. ::: True. $\frac{8+4}{2+1}$ means $\frac{(8+4)}{(2+1)}=\frac{12}{3}=4$, which is why flattening it to $8+4÷2+1$ gives a different expression. See Fractions — numerator and denominator as grouping. Multiplication is placed above addition in the hierarchy purely by arbitrary human choice. ::: False. It follows from meaning: multiplication is repeated addition, so it must be unpacked into its addition before it can be added to anything else. In $2(3+4)$, the "2 next to the bracket" is a special super-strong multiplication that beats a nearby division. ::: False. Implied multiplication is ordinary same-tier multiplication; under strict convention $48÷2(3+4)$ reads left to right as $48÷2=24$, then $24\times 7=168$. There is no secret priority. A power tower like $2^{3^2}$ is read from the bottom up, doing $2^3$ first. ::: False. Exponentiation associates right to left, so the top exponent binds first: $3^2=9$, then $2^9=512$ — not $(2^3{)}^2=64$.

"$8-3+2=8-5=3$, because A comes before S in BODMAS." ::: The error is treating $+$ as higher than $-$. They are one tier: go left to right, $8-3=5$, then $5+2=7$. "$-4^2=16$, since a negative times a negative is positive." ::: There is no second negative here — the base is $4$, not $-4$. The power acts first: $-(4^2)=-16$. "$100÷10÷2=100÷5=20$, because I did the two divisions together." ::: Divisions are same-tier and must go left to right: $100÷10=10$, then $10÷2=5$. Grouping the divisors secretly inserts a bracket that isn't there. "In $6+2[1+(2{)}^2]$ I did $6+2=8$ first because it's on the left." ::: Left-to-right only breaks ties within the same tier. The bracket is a higher tier and must finish before any addition outside it. See Arithmetic operations — addition, subtraction, multiplication, division. "$\sqrt{9+16}=3+4=7$." ::: The root groups everything beneath it, so add first: $\sqrt{25}=5$. The bar is an invisible bracket. "$2+3\times 4=5\times 4=20$." ::: Multiplication is a higher tier than addition, so do $3\times 4=12$ first, then $2+12=14$. "$2^{3^2}=8^2=64$." ::: Exponent chains bind right to left, so $3^2=9$ comes first, giving $2^9=512$, not $(2^3{)}^2$. "To evaluate $[2+(3+1)]\times 5$ I started with $2+3=5$." ::: You must finish the innermost bracket $(3+1)=4$ before touching the outer sum, giving $[2+4]=6$, then $6\times 5=30$.

Why is "innermost bracket first" not just a preference but a logical must? ::: The outer bracket's value depends on the inner one — you cannot plug in a number that hasn't been computed yet. Why do exponents sit above multiplication rather than below it? ::: An exponent is repeated multiplication ($a^3=a\times a\times a$), so it is a shorthand one level up, and shorthands must be unpacked before the operation they compress. See Exponents and powers — laws of indices. Why does the world need a fixed order at all, instead of everyone reading left to right? ::: Pure left-to-right would make $2+3\times 4=20$, breaking the promise that $3\times 4$ means one bundled quantity; a shared tiered grammar guarantees every person and calculator returns the same value. Why does a bracket "beat" every other rule, including exponents? ::: Brackets are a deliberate human override — an explicit instruction to compute a chunk as a finished value first, no matter what operations surround it. Why can't we treat "$-$" in front of a number as automatically part of the number? ::: Because $-3^2$ shows the minus is applied after the power unless a bracket ties it to the base; the sign is an operation, not a fixed label. See Negative numbers and the unary minus sign. Why is left-to-right the tiebreaker for equal-rank operations, rather than right-to-left? ::: It's a convention chosen to match how we read text; the key point is that some fixed rule exists so subtraction and division (which are order-sensitive) come out consistently. Why does a stack of exponents break the left-to-right habit and go right to left instead? ::: Because $a^{b^c}$ is defined as $a$ raised to the whole value $b^c$; the top exponent is itself a number that must be finished before the lower base can use it. See Exponents and powers — laws of indices.

What happens to the rules if an expression has no brackets, powers, or ×÷ at all — just $7-2-1$? ::: Only the bottom tier applies, so read left to right: $7-2=5$, then $5-1=4$. Never $7-(2-1)=6$. What is $x^0$-style edge behaviour worth remembering when a power meets a minus, e.g. $-1^{100}$? ::: The power binds first: $1^{100}=1$, then the minus gives $-1$. Even a huge even exponent doesn't flip the outer sign, because the sign was never inside the base. How do you read a triple power tower $2^{2^3}$? ::: Right to left: $2^3=8$ first, then $2^8=256$. The habit of "left first" is exactly what this edge case overturns. An empty bracket or a bracket with one number, like $5+(3)$ — does the rule still apply? ::: Yes; a lone value in brackets just equals itself, so $(3)=3$ and $5+3=8$. Brackets never require an operation inside them. In $0\times (5+3^2)$, must you still evaluate the bracket even though the product will be zero? ::: Formally the order says evaluate the bracket first; but recognising the whole product is $0$ is a valid shortcut because multiplication by zero is zero regardless of the other factor. The rule and the shortcut agree. What does BODMAS say about a nested pair of the same bracket depth, like $(2+1)+(4-1)$? ::: They are independent and equal priority, so evaluate each in turn (left to right is fine): $3+3=6$. Neither depends on the other. Does a unary minus applied to a whole bracket, $-(3+4)$, distribute automatically? ::: Compute the bracket first to get $-(7)=-7$. You may also distribute the sign, but the order-of-operations reading is "finish the bracket, then negate." See Algebraic expressions — evaluating and substitution. If an expression is genuinely written ambiguously (like the infamous $48÷2(1+2)$), what does BODMAS actually decide? ::: It's not ambiguous under strict convention: implied multiplication is same-tier as division, so left to right gives $48÷2=24$, then $24\times 3=72$. The real fix is to add brackets so no reader has to guess. See Calculator vs mental arithmetic — parsing expressions.