<P>Answer:

The small number written above and to the right of a number is called an \goldD{\text{exponent}}exponentstart color #e07d10, start text, e, X, p, o, n, e, n, t, end text, end color #e07d10. The number underneath the exponent is called the \blueD{\text{base}}basestart color #11accd, start text, b, a, s, e, end text, end color #11accd. In this EXAMPLE, the base is \blueD44start color #11accd, 4, end color #11accd, and the exponent is \goldD33start color #e07d10, 3, end color #e07d10.

Here's an example where the base is \blueD77start color #11accd, 7, end color #11accd, and the exponent is \goldD55start color #e07d10, 5, end color #e07d10:

\blueD7^\goldD57

5

start color #11accd, 7, end color #11accd, start superscript, start color #e07d10, 5, end color #e07d10, end superscript

An exponent tells us to multiply the base by itself that number of times. In our example, \blueD4^\goldD34

3

start color #11accd, 4, end color #11accd, start superscript, start color #e07d10, 3, end color #e07d10, end superscript tells us to multiply the base of \blueD44start color #11accd, 4, end color #11accd by itself \goldD33start color #e07d10, 3, end color #e07d10 times:

\blueD4^\goldD3 =\blueD4 \times \blueD4 \times \blueD44

3

=4×4×4start color #11accd, 4, end color #11accd, start superscript, start color #e07d10, 3, end color #e07d10, end superscript, equals, start color #11accd, 4, end color #11accd, times, start color #11accd, 4, end color #11accd, times, start color #11accd, 4, end color #11accd

Once we write out the multiplication problem, we can easily evaluate the expression. Let's do this for the example we've been working with:

\blueD4^\goldD3 =\blueD4 \times \blueD4 \times \blueD44

3

=4×4×4start color #11accd, 4, end color #11accd, start superscript, start color #e07d10, 3, end color #e07d10, end superscript, equals, start color #11accd, 4, end color #11accd, times, start color #11accd, 4, end color #11accd, times, start color #11accd, 4, end color #11accd

\phantom{\blueD4^\goldD3}= 16 \times 44

3

=16×4empty space, equals, 16, times, 4

\phantom{\blueD4^\goldD3}= 644

3

=64empty space, equals, 64

The main reason we use exponents is because it's a shorter way to write out big numbers. For example, let's say we want to express the following:

\blueD2 \times \blueD2 \times \blueD2 \times \blueD2 \times \blueD2 \times \blueD22×2×2×2×2×2start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd

That's REALLY long to write. My hands HURT just from typing it! Instead we can see that \blueD22start color #11accd, 2, end color #11accd is multiplied by itself \goldD66start color #e07d10, 6, end color #e07d10 times. This means we can write the same thing with \blueD22start color #11accd, 2, end color #11accd as the base and \goldD66start color #e07d10, 6, end color #e07d10 as the exponent:

\blueD2 \times \blueD2 \times \blueD2 \times \blueD2 \times \blueD2 \times \blueD2 = \blueD2^\goldD62×2×2×2×2×2=2

6