Topic 11 of 12

Exponents and inequalities

Step 8 · Ratios and negative numbers topics

Write repeated multiplication with exponents, evaluate expressions with them using the order of operations, and use the symbols < and > to describe which numbers make a statement true.

Exponents

An exponent says how many times to multiply a number by itself. In $2^3$ , the base is 2 and the exponent is 3, so the 2 is used three times:

2^3 = 2 \times 2 \times 2 = 8

It is not $2 \times 3 = 6$ . The exponent counts how many copies of the base are multiplied, so it is not the number you multiply by. Exponents 2 and 3 have special names: $5^2$ is read "five squared" and $5^3$ "five cubed", because squares and cubes are how we measure area and volume.

Power	Meaning	Value
$5^2$	$5 \times 5$	25
$10^3$	$10 \times 10 \times 10$	1000
$3^4$	$3 \times 3 \times 3 \times 3$	81
$1.5^2$	$1.5 \times 1.5$	2.25

Exponents in the order of operations

Work in this order: grouping symbols first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

Example: Exponent before multiplication

Find $3 + 2^2 \times 5$ .

Exponents first: $2^2 = 4$ , so the expression is $3 + 4 \times 5$ .
Multiply: $4 \times 5 = 20$ .
Add: $3 + 20 = 23$ .

Answer23

Example: Parentheses change things

Find $(3 + 2)^2$ .

Do the parentheses first: $3 + 2 = 5$ .
Then the exponent: $5^2 = 25$ .

Answer25, not the 13 you get from $3^2 + 2^2$ .

Inequalities

An inequality compares two quantities that may not be equal. The symbols are:

$<$ means less than, and $>$ means greater than.
$\leq$ means less than or equal to, and $\geq$ means greater than or equal to.

A number is a solution of an inequality if the statement is true when you put it in. For $x + 2 > 8$ , the number 7 is a solution, because $7 + 2 = 9$ and $9 > 8$ . The number 6 is not, because $6 + 2 = 8$ and 8 is not greater than 8.

Example: Testing a value

Is 12 a solution of $3x \leq 30$ ?

Put in $x = 12$ : $3 \times 12 = 36$ .
Is $36 \leq 30$ ? No.

AnswerNo, 12 is not a solution.

Many solutions

Most inequalities have more than one solution, often infinitely many. All the numbers greater than 3 make $x > 3$ true: 3.1, 4, 100, and so on. On a number line the solutions are shaded, and the end point 3 is marked as not included.

A number line from 0 to 8 with 3 marked. The solutions of x > 3 are every number to the right of 3, such as 4, 5 and 7.5. The number 3 itself is not a solution.

Writing inequalities

Words become inequalities. "At most 12 passengers" means the number of passengers $p$ can be 12 or less, so $p \leq 12$ . "More than 5 years old" means $a > 5$ .

Try it yourself

Work each one on paper first, then open the answer to check.

Find $4^3$ .
Show answer
64, because $4 \times 4 \times 4 = 64$ .
Find $2^5$ .
Show answer
32, because $2 \times 2 \times 2 \times 2 \times 2 = 32$ .
Find $20 - 3^2$ .
Show answer
11, because $3^2 = 9$ and $20 - 9 = 11$ .
Is 5 a solution of $x + 4 \geq 9$ ?
Show answer
Yes, because $5 + 4 = 9$ and $9 \geq 9$ is true.
Write "a number $n$ is less than 10" as an inequality.
Show answer
$n < 10$
Which whole numbers make $2n < 9$ true?
Show answer
0, 1, 2, 3 and 4. For 4, $2 \times 4 = 8 < 9$ . For 5, $2 \times 5 = 10$ , which is not less than 9.