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.