You know the order of operations for parentheses, multiplying, dividing, adding and subtracting. Exponents fit in right after the grouping symbols.
Idea clave
Work in this order: (1) grouping symbols, (2) exponents, (3) multiply and divide from left to right, (4) add and subtract from left to right.
Por qué funciona
An exponent is a product that has been bundled up: $2^2$ is $2 \times 2$. In $3 \times 2^2$ the exponent belongs only to the 2, so that bundle is finished first: $3 \times (2 \times 2) = 12$. If we wanted $(3 \times 2)^2 = 36$, we would have written the parentheses. Everyone using the same order means an expression has only one value.
| Step | Look for | Example |
|---|---|---|
| 1 | parentheses and other grouping | $(2 + 4)^2 \to 6^2$ |
| 2 | exponents | $6^2 \to 36$ |
| 3 | $\times$ and $\div$, left to right | $36 \div 4 \times 2 \to 9 \times 2$ |
| 4 | $+$ and $-$, left to right | $18 - 1 + 3 \to 17 + 3$ |
Ejemplo resuelto: An exponent and a product
Find $5 + 3^2 \times 2 - 4$.
- Exponent: $3^2 = 9$. Now $5 + 9 \times 2 - 4$.
- Multiply: $9 \times 2 = 18$. Now $5 + 18 - 4$.
- Left to right: $23 - 4 = 19$.
Respuesta: $19$
Ejemplo resuelto: Parentheses first
Find $(2 + 4)^2 \div 4 - 1$.
- Parentheses: $2 + 4 = 6$.
- Exponent: $6^2 = 36$.
- Divide: $36 \div 4 = 9$. Subtract: $9 - 1 = 8$.
Respuesta: $8$
Ejemplo resuelto: Divide and multiply left to right
Find $48 \div 2^3 \times 3$.
- Exponent: $2^3 = 8$. Now $48 \div 8 \times 3$.
- Left to right: $48 \div 8 = 6$, then $6 \times 3 = 18$.
Respuesta: $18$
Error común
Multiplying before the exponent: $3 \times 2^2$ worked as $6^2 = 36$. Quick check: an exponent sticks to the one number (or parentheses) right before it. Here that is the 2, so square the 2 first: $3 \times 4 = 12$.