Pre-Algebra

Exponents and Powers

Lesson

An exponent is a shortcut for repeated multiplication. $a^n$ means: multiply $a$ by itself $n$ times.

$a$ is the base.
$n$ is the exponent (or power).
We say $a^2$ as “a squared” and $a^3$ as “a cubed”.

Worked example 1

3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81

Worked example 2

5^2 = 5 \cdot 5 = 25

Two special cases

Any nonzero number to the 0 power is 1: $7^0 = 1$ .
Any number to the 1 power is itself: $9^1 = 9$ .

Order of operations: exponents come before multiplication, division, addition, and subtraction (but after parentheses).

Worked example 3 — order matters

2 + 3^2 = 2 + 9 = 11

Compute the exponent first, then add. NOT $5^2$ .

Watch the parentheses

$(-3)^2 = 9$ — the base is $-3$ , and $(-3)(-3) = 9$ .

But $-3^2 = -9$ — the base is just $3$ , and the negative sticks on at the end.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

2^3

Problem 2

5^2

Problem 3

4^2

Problem 4

10^2

Practice

Standard problems matching the lesson.

Problem 5

3^3

Problem 6

6^2

Problem 7

2^4

Problem 8

10^3

Problem 9

7^0

Problem 10

9^1

Problem 11

1^{15}

Problem 12

0^5

Problem 13

4 + 2^3

Problem 14

5^2 - 4^2

Problem 15

(-3)^2

Problem 16

-3^2

Problem 17

8^2

Problem 18

5^3

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

2^5

Problem 20

3^4

Problem 21

(-2)^3

Problem 22

(-4)^2 - 5

Problem 23

2^3 \cdot 3^2

Problem 24

10^2 - 6^2

Problem 25

2^6

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