Order of Operations with Exponents, Fractions and Decimals | Grade 6 Math | LevelKite

A numerical expression combines numbers with operations such as addition and multiplication. It has no equals sign and no letters. One example is $2 + 3 \times 4$ . To evaluate an expression means to find the one number it equals. That number is the value of the expression.

Try it with $2 + 3 \times 4$ . If you add first, you get $5 \times 4 = 20$ . If you multiply first, you get $2 + 12 = 14$ . An expression must have just one value. So everyone follows the same agreed order. Its name is the order of operations. You will learn it here and use it with whole numbers, fractions and decimals.

The agreed order fits what the symbols mean. Multiplication can be a short way to write repeated addition. Here, $3 \times 4$ means three copies of $4$ added together: $4 + 4 + 4$ . So $2 + 3 \times 4$ means $2 + 4 + 4 + 4$ , which is $14$ . That is why the agreed order puts multiplication before addition.

Sometimes you want to add first. Then you write parentheses, the curved marks around part of an expression. Parentheses are grouping symbols: they tell you to treat everything inside them as one number. So $(2 + 3) \times 4$ means $5 \times 4$ , which is $20$ .

Now look at the power $2^4$ . Its base is $2$ and its exponent is $4$ . You already know what this means: the exponent $4$ tells you to use the base as a factor four times. A factor is a number that you multiply. So this power is a short form too: $2^4$ is a short way to write $2 \times 2 \times 2 \times 2$ .

An exponent belongs only to its base. In $3 \times 2^4$ , the base is $2$ , not $3 \times 2$ . So you find the power first. Then you multiply by $3$ .

3 \times 2^4 = 3 \times (2 \times 2 \times 2 \times 2) = 3 \times 16 = 48

Key concept

The order of operations

To evaluate a numerical expression, do the operations in this order:

Parentheses. Work out what is inside them first. Inside, use this same order.
Exponents. Find the value of each power.
Multiplication and division. They share this step. Do them in the order they appear, from left to right.
Addition and subtraction. They share this step. Do them in the order they appear, from left to right.

Skip a step when the expression has nothing for it. After each step, rewrite the expression and put the value in place of the part you just worked out. The new line has the same value as the old one, because you swapped a part for a number equal to it.

Worked example

The full order with whole numbers

Evaluate $7 + 40 \div 2^3 \times (9 - 6)$ .

Start inside the parentheses, because they group 9 minus 6 into one number. That number is 3.
$7 + 40 \div 2^3 \times (9 - 6) = 7 + 40 \div 2^3 \times 3$
Find the power next. The base is 2 and the exponent is 3, so the power means 2 times 2 times 2, which is 8.
$7 + 40 \div 2^3 \times 3 = 7 + 40 \div 8 \times 3$
Multiplication and division share a step, so read from the left. The division comes first: 40 divided by 8 is 5.
$7 + 40 \div 8 \times 3 = 7 + 5 \times 3$
Now multiply: 5 times 3 is 15. The addition sits at the far left, but it waits, because multiplication comes before addition.
$7 + 5 \times 3 = 7 + 15$
Only the addition is left: 7 plus 15 is 22.
$7 + 15 = 22$

Answer

The value of $7 + 40 \div 2^3 \times (9 - 6)$ is $22$ .

The order of operations is about the operations, not about the kind of number. So when an expression has fractions or decimals, you follow the same four steps in the same order. Only the small calculations inside each step change. For those, you use the fraction and decimal arithmetic you already know. Write each small calculation down, so you can check it.

Before you start, find the base of each power. If a number comes just before the exponent, that one number is the base. If parentheses end just before the exponent, the base is everything inside them. This matters a lot with fractions. To raise a whole fraction to a power, you put parentheses around it, as in $\left(\frac{2}{3}\right)^2$ . Compare each pair of rows in the table.

How parentheses change the base of a power

Expression	Base of the power	Value
$5 + 3^2$	$3$	$5 + 9 = 14$
$(5 + 3)^2$	$5 + 3$ , which is $8$	$8^2 = 64$
$2 \times 0.3^2$	$0.3$	$2 \times 0.09 = 0.18$
$(2 \times 0.3)^2$	$2 \times 0.3$ , which is $0.6$	$0.6^2 = 0.36$
$\frac{2^2}{3}$	$2$ , the top number only	$\frac{4}{3}$ , because $2^2 = 4$
$\left(\frac{2}{3}\right)^2$	$\frac{2}{3}$ , the whole fraction	$\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$

Worked example

Decimals with parentheses

Evaluate $1.5 + 8 \times (0.7 - 0.2)^2$ .

Start inside the parentheses: 0.7 minus 0.2 is 0.5. The parentheses end just before the exponent, so 0.5 becomes the base of the power.
$1.5 + 8 \times (0.7 - 0.2)^2 = 1.5 + 8 \times 0.5^2$
Find the power next. It means 0.5 times 0.5. Five tenths times five tenths is 25 hundredths, which is 0.25.
$1.5 + 8 \times 0.5^2 = 1.5 + 8 \times 0.25$
Multiply before you add. 8 times 25 hundredths is 200 hundredths, which is 2.
$1.5 + 8 \times 0.25 = 1.5 + 2$
Only the addition is left: 1.5 plus 2 is 3.5.
$1.5 + 2 = 3.5$

Answer

The value of $1.5 + 8 \times (0.7 - 0.2)^2$ is $3.5$ .

Worked example

Fractions with a fraction base

Evaluate $\frac{3}{4} \times \left(\frac{5}{6} - \frac{1}{3}\right) + \left(\frac{1}{2}\right)^3$ .

Start with the parentheses that hold an operation. To subtract, both fractions need the same denominator, which is the bottom number. Each third is the same size as two sixths, so one third equals two sixths. Subtract to get three sixths. Three sixths is half of six sixths, so it equals one half.
$\frac{5}{6} - \frac{1}{3} = \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$
The other parentheses hold one number and no operation. They show that the whole fraction is the base. The exponent is 3, so use the fraction as a factor three times. Multiply the top numbers, then multiply the bottom numbers.
$\left(\frac{1}{2}\right)^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1 \times 1 \times 1}{2 \times 2 \times 2} = \frac{1}{8}$
Rewrite the expression with these two values in place. Its value stays the same, because each part you replaced equals the number you put in.
$\frac{3}{4} \times \frac{1}{2} + \frac{1}{8}$
Multiply before you add. Multiply the top numbers, then multiply the bottom numbers.
$\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}$
Only the addition is left. The denominators already match, so add the top numbers to get four eighths. Four eighths is half of eight eighths, so it equals one half.
$\frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2}$

Answer

The value of $\frac{3}{4} \times \left(\frac{5}{6} - \frac{1}{3}\right) + \left(\frac{1}{2}\right)^3$ is $\frac{1}{2}$ .

The takeaway: an expression has just one value because everyone evaluates it in the same order. Parentheses come first. Exponents come next. Then you multiply and divide from left to right. Last, you add and subtract from left to right. An exponent applies only to its base, so find each base before you start. Fractions and decimals do not change this order. They only change the arithmetic inside each step.

Next, you will write expressions for word problems, compare the values of expressions, and spot mistakes that people make when they evaluate expressions with exponents.