Skill: Write expressions for repeated operations

A repeated operation can often be written more briefly.

• Repeated addition of the same amount becomes multiplication.
• Repeated multiplication of the same factor becomes an exponent.

Worked example

A box contains $(x+2)$ pencils. There are $3$ identical boxes. Write an expression for the total number of pencils.

Step 1: Write the repeated addition.

Each box has $(x+2)$ pencils, so add that amount $3$ times:

$$(x+2)+(x+2)+(x+2)$$

Step 2: Replace the repeated addition with multiplication.

There are $3$ groups of $(x+2)$, so write:

$$\boxed{3(x+2)}$$

The parentheses are important because the whole quantity $(x+2)$ is repeated.

• $(3(x+2))$ means $3$ groups of $(x+2)$.
• $(3x)$ means $3$ groups of only $x$.
• $(x+3)$ means $3$ is added to $x$.

For repeated multiplication, use an exponent. For example,

$$2\cdot2\cdot2\cdot2=2^4$$

The exponent $4$ tells how many times the factor $2$ is used.