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`.

Answer: 64.

Next example.

Example 2. Find `3^4`.

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

Answer: 81.

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

Example 3. Find `(-3)^2`.

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

Answer: 9.

Next example.

Example 4. Find `(-5)^3`.

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

Answer: -125.

Last example.

Example 5. Find `(-1461)^0`.

`(-1461)^0=1`.

Answer: 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`.

Answer: 9.

Next example.

Exercise 2. Find `1^15`.

`1^(15)=1`.

Answer: 1.

Next exercise.

Exercise 3. Find `2^3`.

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

Answer: 8.

Next exercise.

Exercise 4. Find `(-3)^3`.

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

Answer: -27.

A couple more.

Exercise 5. Find `(-5)^4`.

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

Answer: 625.

Exercise 6. Find `-5^4`.

`-5^4=-5*5*5*5=-625`.

Answer: -625.

Exercise 7. Find `-(-2)^6`.

`-(-2)^6=-(-2)*(-2)*(-2)*(-2)*(-2)*(-2)=-64`.

Answer: -64.