# Exponents and Integers

Let's learn about positive integer exponents.

We already know how to multiply integers.

Indeed, you've learned, that 2*2=4, 2*2*2=8, 2*2*2*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*2*2*2*2*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*0*0*0*0*0*0*0=0.
• One raised to any power is one: 1^a=1. For example, 1^5=1*1*1*1*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*2*2*2*2=32 and 5^2=5*5=25.

Let's go through a couple of examples.

Example 1 . Find 4^3.

4^3=4*4*4=64.

Next example.

Example 2. Find 3^4.

3^4=3*3*3*3=81.

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

Example 3. Find (-3)^2.

(-3)^2=(-3)*(-3)=9.

Next example.

Example 4. Find (-5)^3.

(-5)^3=(-5)*(-5)*(-5)=25*(-5)=-125.

Last example.

Example 5. Find (-1461)^0.

(-1461)^0=1.

Word of caution: pay attention to parenthesis and minuses:

• (-4)^2=(-4)*(-4)=16
• -4^2=-(4*4)=-16

Now, take pen and paper and solve following problems.

Exercise 1. Find 3^2.

3^2=3*3=9.

Next example.

Exercise 2. Find 1^15.

1^(15)=1.

Next exercise.

Exercise 3. Find 2^3.

2^3=2*2*2=8.

Next exercise.

Exercise 4. Find (-3)^3.

(-3)^3=(-3)*(-3)*(-3)=9*(-3)=-27.

A couple more.

Exercise 5. Find (-5)^4.

(-5)^4=(-5)*(-5)*(-5)*(-5)=25*(-5)*(-5)=-125*(-5)=625.

Exercise 6. Find -5^4.
-5^4=-5*5*5*5=-625.
Exercise 7. Find -(-2)^6.
-(-2)^6=-(-2)*(-2)*(-2)*(-2)*(-2)*(-2)=-64.