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.