Why is y × y × y equal to y³ and not 3y?

Because y × y × y is three y's multiplied together, and repeated multiplication builds a power. The index (3) counts how many times the base (y) is multiplied by itself: y × y × y = y³. Writing 3y means three y's added — that is y + y + y, not y × y × y. Check with y = 2: 3y = 6 but y × y × y = 8, so they are different expressions.

Why is y + y + y equal to 3y and not y³?

Because y + y + y is three y's added together, and repeated addition builds a coefficient. The coefficient (3) counts how many copies of y are being summed: y + y + y = 3y. Writing y³ means y multiplied by itself three times — that is y × y × y, not y + y + y. The operation in the middle decides the rule: + gives a coefficient, × gives a power.

Why is 4 × 2c equal to 8c and not 6c?

Because 4 × 2c means multiply 4 by the entire term 2c. The numbers multiply: 4 × 2 = 8. The variable c is unchanged. So 4 × 2c = 8c. Writing 6c adds the coefficients (4 + 2 = 6) instead of multiplying them — the sign is ×, so the numbers must be multiplied.

Why is 2m ÷ m equal to 2 and not 2m?

Because dividing 2m by m cancels the variable. The coefficient divides: 2 ÷ 1 = 2. The variable divides: m ÷ m = 1. So 2m ÷ m = 2 × 1 = 2. Writing 2m keeps the m when it has already been divided out. Think of it as a fraction: 2m/m — the m in the numerator and denominator cancel, leaving 2.

GCSE Maths Foundation · AQA · Algebraic manipulation

Coefficient vs index: why y × y × y is y³, not 3y

The most common simplification slip in algebraic manipulation: students count three $y$ s in $y \times y \times y$ and write $3y$ , as if the operation were addition. AQA Foundation examiners report this trap on nearly every paper that includes a simplify question. The mirror error is equally common — faced with $y + y + y$ , students write $y^3$ , applying the multiplication model to an addition.

The thirty-second fix: read the sign in the middle first. A $+$ sign builds a coefficient ( $y + y + y = 3y$ ). A $\times$ sign builds an index ( $y \times y \times y = y^3$ ). Same letter, same count of three, opposite operation, opposite result.

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How to spot it in your own work

You wrote $3y$ for $y \times y \times y$ by counting the three $y$ s instead of reading the $\times$ sign.
You wrote $y^3$ for $y + y + y$ by applying the multiplication rule to an addition.
You simplified $4 \times 2c$ as $6c$ by adding the coefficients (4 + 2) instead of multiplying them (4 × 2 = 8).
You wrote $2m$ for $2m \div m$ without cancelling the variable.

An exam question that triggers it

Here is the canonical AQA Foundation trigger, the shape of Jun24 Paper 2 Q2d:

Simplify.

y \times y \times y

[1 mark]

The misconception answer is $3y$ , reached by counting the three $y$ s and treating the count as a coefficient. But the sign between them is $\times$ , not $+$ .

The correct answer is $y^3$ . Three $y$ s multiplied together build a power of 3. Check at $y = 2$ : $3y = 6$ but $y \times y \times y = 8$ — they are different.

Why students fall for this

When a student sees $y \times y \times y$ , the most automatic response is to count the letters: three $y$ s, so the answer must involve the number 3. Putting 3 in front of the letter — writing $3y$ — is the default move because it mirrors the way coefficients are written everywhere else in early algebra. The student is applying the right logic (count the letters) to the wrong operation.

The mirror error ( $y + y + y = y^3$ ) happens because students who have recently met index notation try to use it: seeing three identical letters, they reach for the power form. Both traps share the same root — the operation sign in the middle is being ignored.

The coefficient multiplication trap ( $4 \times 2c = 6c$ ) is the same habit one layer up: the student adds the numbers in front (4 + 2 = 6) instead of multiplying them, perhaps because adding feels like the natural way to combine numbers alongside variables.

The fix: Read the sign first: + makes a coefficient, × makes a power

Adding identical copies of a variable builds a coefficient. $y + y + y$ — three $y$ s added — the coefficient counts the copies:

y + y + y = 3y

Multiplying identical copies of a variable builds a power. $y \times y \times y$ — three $y$ s multiplied — the index counts the multiplications:

y \times y \times y = y^3

When × sits between a number and a term, multiply the numbers. $4 \times 2c$ : multiply the coefficients ( $4 \times 2 = 8$ ), keep the variable ( $c$ ) →

4 \times 2c = 8c

When ÷ cancels the variable, the result is a plain number. $2m \div m$ : the coefficient divides ( $2 \div 1 = 2$ ), the variable cancels ( $m \div m = 1$ ) →

2m \div m = 2

Worked example

Simplify each expression. [1 mark each]

$y \times y \times y$
Read the sign: ×. Multiplying copies builds a power. Count the multiplications: three. So the index is 3.
$y \times y \times y = y^3$
Trap: $3y$ — that counts the letters into a coefficient instead of reading the × sign.
$y + y + y$
Read the sign: +. Adding copies builds a coefficient. Count the copies: three. So the coefficient is 3.
$y + y + y = 3y$
Trap: $y^3$ — that applies the power rule to an addition.
$4 \times 2c$
Multiply the numbers, keep the variable. $4 \times 2 = 8$ , variable $c$ unchanged.
$4 \times 2c = 8c$
Trap: $6c$ — that adds 4 + 2 instead of multiplying.
$2m \div m$
Cancel the variable. Write as a fraction: $\frac{2m}{m}$ . The $m$ in numerator and denominator cancel.
$2m \div m = \frac{2m}{m} = 2$
Trap: $2m$ — that leaves the variable in when it has already cancelled.

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Common questions

How do I tell whether to write a coefficient or a power?: Read the sign between the letters. If the letters are added ( $+$ ), count them and put the count in front as a coefficient: $y + y + y = 3y$ . If the letters are multiplied ( $\times$ ), count the multiplications and write the count as a superscript power: $y \times y \times y = y^3$ . The sign decides everything.
Why isn't $y \times y \times y$ the same as $3y$ ?: Substitute a number to check. At $y = 2$ : $3y = 3 \times 2 = 6$ , but $y \times y \times y = 2 \times 2 \times 2 = 8$ . They give different values, so they are different expressions. The $\times$ sign means multiply the $y$ s together — that is a power, not repeated addition.
Why is 4 × 2c equal to 8c and not 6c?: The operation between 4 and 2c is multiplication, so the numbers multiply: $4 \times 2 = 8$ . The variable $c$ is unchanged. Result: $8c$ . Writing $6c$ adds the coefficients (4 + 2 = 6) — but the sign is ×, not +, so addition is the wrong operation.
How does 2m ÷ m simplify to 2?: Write it as a fraction: $\frac{2m}{m}$ . The $m$ on top and the $m$ on the bottom are the same, so they cancel — just as $\frac{5}{5} = 1$ . That leaves $\frac{2}{1} = 2$ . The variable has gone; the answer is a plain number.

Related misconceptions

Combining unlike terms: why 3x + 2y stays as 3x + 2yThe same need to read the letter carefully — unlike terms cannot be collected, no matter how similar they look.
Fraction of an amount: why 2/5 of 1020 is 408, not 40Another reading error: treating an instruction as a label instead of an operation to carry out.
Function machines: why 12 → [−4] → [×5] is 40, not −8The same need to read the operation sign carefully before deciding what to do.

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