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 ${{4}}^{{3}}$.

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

Next example.

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

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

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

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

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

Next example.

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

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

Last example.

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

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

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 ${{3}}^{{2}}$.

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

Next example.

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

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

Next exercise.

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

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

Next exercise.

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

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

A couple more.

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

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

Exercise 6. Find $-{{5}}^{{4}}$.
$-{{5}}^{{4}}=-{5}\cdot{5}\cdot{5}\cdot{5}=-{625}$.
Exercise 7. Find $-{{\left(-{2}\right)}}^{{6}}$.
$-{{\left(-{2}\right)}}^{{6}}=-{\left(-{2}\right)}\cdot{\left(-{2}\right)}\cdot{\left(-{2}\right)}\cdot{\left(-{2}\right)}\cdot{\left(-{2}\right)}\cdot{\left(-{2}\right)}=-{64}$.