Exponents and Integers

Let's learn about positive integer exponents.

We already know how to multiply integers.

Indeed, you've learned, that $$${2}\cdot{2}={4}$$$, $$${2}\cdot{2}\cdot{2}={8}$$$, $$${2}\cdot{2}\cdot{2}\cdot{2}={16}$$$.

But what if you want to multiply same number certain number of times?

Suppose, we multiply 2 by itself six times. We, of course, can write it in following way: $$${2}\cdot{2}\cdot{2}\cdot{2}\cdot{2}\cdot{2}={64}$$$.

But this notation is too long, so there is special notation: we write it as $$${{2}}^{{6}}={64}$$$.

Raising $$${a}$$$ to $$${b}$$$-th power is $$$\color{purple}{a^b=\underbrace{a\cdot a\cdot a\cdot a\cdot...\cdot a}_{b}}$$$.

$$${a}$$$ is called base, $$${b}$$$ is exponent (power).

For now we assume that $$${b}$$$ is positive integer. We will see later what it means, when $$${b}$$$ is not positive integer.

In other words raising to power (exponentiation) tells us how many times to use number in multiplication.

There are nice facts about exponents.

  • Zero raised to any non-zero power is zero: $$${{0}}^{{a}}={0}$$$. For example, $$${{0}}^{{8}}={0}\cdot{0}\cdot{0}\cdot{0}\cdot{0}\cdot{0}\cdot{0}\cdot{0}={0}$$$.
  • One raised to any power is one: $$${{1}}^{{a}}={1}$$$. For example, $$${{1}}^{{5}}={1}\cdot{1}\cdot{1}\cdot{1}\cdot{1}={1}$$$.
  • Any number raised to the zero power is 1: $$${{a}}^{{0}}={1}$$$. For example, $$${{15}}^{{0}}={1}$$$.
  • Any number raised to the first power is number itself: $$${{a}}^{{1}}={a}$$$. For example, $$${{357}}^{{1}}={357}$$$.

Word of Caution. There is huge difference between $$${{a}}^{{b}}$$$ and $$${{b}}^{{a}}$$$.

For example, $$${{2}}^{{5}}={2}\cdot{2}\cdot{2}\cdot{2}\cdot{2}={32}$$$ and $$${{5}}^{{2}}={5}\cdot{5}={25}$$$.

Let's go through a couple of examples.

Example 1. Find $$${{4}}^{{3}}$$$.

$$${{4}}^{{3}}={4}\cdot{4}\cdot{4}={64}$$$.

Answer: 64.

Next example.

Example 2. Find $$${{3}}^{{4}}$$$.

$$${{3}}^{{4}}={3}\cdot{3}\cdot{3}\cdot{3}={81}$$$.

Answer: 81.

Now, let's see how to deal with negative integers.

Example 3. Find $$${{\left(-{3}\right)}}^{{2}}$$$.

$$${{\left(-{3}\right)}}^{{2}}={\left(-{3}\right)}\cdot{\left(-{3}\right)}={9}$$$.

Answer: 9.

Next example.

Example 4. Find $$${{\left(-{5}\right)}}^{{3}}$$$.

$$${{\left(-{5}\right)}}^{{3}}={\left(-{5}\right)}\cdot{\left(-{5}\right)}\cdot{\left(-{5}\right)}={25}\cdot{\left(-{5}\right)}=-{125}$$$.

Answer: -125.

Last example.

Example 5. Find $$${{\left(-{1461}\right)}}^{{0}}$$$.

$$${{\left(-{1461}\right)}}^{{0}}={1}$$$.

Answer: 1.

Word of caution: pay attention to parenthesis and minuses:

  • $$${{\left(-{4}\right)}}^{{2}}={\left(-{4}\right)}\cdot{\left(-{4}\right)}={16}$$$
  • $$$-{{4}}^{{2}}=-{\left({4}\cdot{4}\right)}=-{16}$$$

Now, take pen and paper and solve following problems.

Exercise 1. Find $$${{3}}^{{2}}$$$.

$$${{3}}^{{2}}={3}\cdot{3}={9}$$$.

Answer: 9.

Next example.

Exercise 2. Find $$${{1}}^{{15}}$$$.

$$${{1}}^{{{15}}}={1}$$$.

Answer: 1.

Next exercise.

Exercise 3. Find $$${{2}}^{{3}}$$$.

$$${{2}}^{{3}}={2}\cdot{2}\cdot{2}={8}$$$.

Answer: 8.

Next exercise.

Exercise 4. Find $$${{\left(-{3}\right)}}^{{3}}$$$.

$$${{\left(-{3}\right)}}^{{3}}={\left(-{3}\right)}\cdot{\left(-{3}\right)}\cdot{\left(-{3}\right)}={9}\cdot{\left(-{3}\right)}=-{27}$$$.

Answer: -27.

A couple more.

Exercise 5. Find $$${{\left(-{5}\right)}}^{{4}}$$$.

$$${{\left(-{5}\right)}}^{{4}}={\left(-{5}\right)}\cdot{\left(-{5}\right)}\cdot{\left(-{5}\right)}\cdot{\left(-{5}\right)}={25}\cdot{\left(-{5}\right)}\cdot{\left(-{5}\right)}=-{125}\cdot{\left(-{5}\right)}={625}$$$.

Answer: 625.

Exercise 6. Find $$$-{{5}}^{{4}}$$$.

$$$-{{5}}^{{4}}=-{5}\cdot{5}\cdot{5}\cdot{5}=-{625}$$$.

Answer: -625.

Exercise 7. Find $$$-{{\left(-{2}\right)}}^{{6}}$$$.

$$$-{{\left(-{2}\right)}}^{{6}}=-{\left(-{2}\right)}\cdot{\left(-{2}\right)}\cdot{\left(-{2}\right)}\cdot{\left(-{2}\right)}\cdot{\left(-{2}\right)}\cdot{\left(-{2}\right)}=-{64}$$$.

Answer: -64.