Order of Operations: Additional Practice (Set 2)

The bar groups the numerator and denominator, so settle each first.

$\frac{2^3 + 4^2}{6 - 2} = \frac{8 + 16}{4} = \frac{24}{4}$

Then divide.

$\frac{24}{4} = 6$

Start at the innermost group and move outward; the exponent applies once the parentheses settle.

$2 \times \{\,12 - [\,3 + (8 - 6)^2\,]\,\} = 2 \times \{\,12 - [\,3 + 2^2\,]\,\} = 2 \times \{\,12 - [\,3 + 4\,]\,\}$

Finish the bracket, then the brace, then multiply.

$2 \times (12 - 7) = 2 \times 5 = 10$

Settle the numerator and the denominator separately.

$\frac{(5 - 1)^2}{2^3} = \frac{4^2}{8} = \frac{16}{8}$

Then divide.

$\frac{16}{8} = 2$

The bar groups the numerator and denominator, so settle the fraction; handle the parenthesized difference and its product separately.

$\frac{5^2 + 5}{2 + 3} + 2 \times (7 - 4) = \frac{25 + 5}{5} + 2 \times 3 = \frac{30}{5} + 6$

Divide, then add.

$6 + 6 = 12$

Work from the innermost group outward.

$10 - [\,2 + 3 \times (4 - 2)\,] = 10 - [\,2 + 3 \times 2\,]$

Inside the bracket, multiply before adding.

$10 - [\,2 + 6\,] = 10 - 8 = 2$

Work the innermost group, apply its exponent, then the standalone power, then the product, then subtract.

$2^4 - 3 \times [\,(5 - 2)^2 - 5\,] = 16 - 3 \times [\,3^2 - 5\,] = 16 - 3 \times [\,9 - 5\,]$

Finish the bracket, multiply, then subtract.

$16 - 3 \times 4 = 16 - 12 = 4$

The fraction bar groups the whole numerator, so settle the top before dividing.

$\frac{16 + 9}{5} + 3 = \frac{25}{5} + 3 = 5 + 3$

Then add the $+3$, which sits outside the fraction.

$5 + 3 = 8$

Dividing by $5 + 3$ instead of just $5$ gives $\dfrac{25}{8}$; stopping at $\dfrac{25}{5}$ and forgetting the $+3$ gives $5$; and ignoring the bar entirely to add $16 + 9 + 3$ gives $28$.

Square first, then the division and multiplication left to right, then the addition and subtraction left to right.

$6 + 24 \div 2^2 \times 3 - 1 = 6 + 24 \div 4 \times 3 - 1 = 6 + 6 \times 3 - 1$

Finish the middle tier, then the bottom tier.

$6 + 18 - 1 = 24 - 1 = 23$

Work from the innermost group outward.

$\{\,[\,(2 + 1) \times 3\,] - 4\,\} \times 2 = \{\,[\,3 \times 3\,] - 4\,\} \times 2 = (9 - 4) \times 2$

Finish the brace, then multiply.

$5 \times 2 = 10$

Division and multiplication share one tier, left to right; then add.

$6 + 12 \div 2 \times 3 = 6 + 6 \times 3$

The $12 \div 2$ comes first because it is leftmost, then the multiplication, then the addition.

$6 + 18 = 24$

Evaluate both exponents first, then the multiplication and division, then subtract.

$2^3 \times 3 - 4^2 \div 8 = 8 \times 3 - 16 \div 8 = 24 - 2$

Then subtract.

$24 - 2 = 22$

Settle the group and the exponent, then the multiplication and division, then subtract.

$5 \times 2^3 - (7 + 5) \div 4 = 5 \times 8 - 12 \div 4 = 40 - 3$

Then subtract.

$40 - 3 = 37$