Exponentiation is a mathematical operation that represents repeated multiplication. The exponent $n$ in the expression $b^n$ represents the number of times the base $b$ is multiplied by itself. 

For example, the expression $b^3$ represents $b \cdot b \cdot b$. Here, the exponent is 3, and the expression can be read in any of the following ways: 

• $b$ raised to the 3rd power
• $b$ raised to the power of $3$
• $b$ raised by the exponent of $3$

Some exponents have their own unique pronunciations. For example, $b^2$ is usually read as "$b$ squared" and $b^3$ as "$b$ cubed." 

Exponentiation is used frequently in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.

Positive Integer Exponents

Now that we understand the basic idea, let's practice simplifying some exponential expressions. 

Example 1

Let's look at an exponential expression with 2 as the base and 3 as the exponent:

$2^3$

This means that the base 2 gets multiplied by itself 3 times:

$2^3 = 2 \cdot 2 \cdot 2 = 8$

Example 2

Let's look at another exponential expression, this time with 3 as the base and 5 as the exponent:

$3^5$

This means that the base 3 gets multiplied by itself 5 times:

$3^5 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 243$

Exponents of 0 and 1

Any number raised by the exponent $1$ is the number itself. That is to say, $b^1=b$.

Any nonzero number raised by the exponent 0 is 1. That is to say, $b^0=1$. For example, $4^0=1$.