\[\begin{align*} \boxed{6^3 = 6 \times 6 \times 6 = 216} \end{align*}\]

Introduction

The expression \( 6^3 \) is an example of exponential notation, where:
– \( 6 \) is the base, which is the number being multiplied.
– \( 3 \) is the exponent, which tells us how many times to multiply the base by itself.

In other words, \( 6^3 \) means “multiply 6 by itself 3 times.”

Step-by-Step Solution

To solve \( 6^3 \), we can expand it as follows:

\[\begin{align*} 6^3 = 6 \times 6 \times 6 \end{align*}\]

Now, let’s multiply each pair step-by-step:

1. Start with the first two factors: \( 6 \times 6 = 36 \)
2. Multiply the result by the last 6: \( 36 \times 6 = 216 \)

Thus:

\[\begin{align*} 6^3 = 216 \end{align*}\]

Alternatively, if we know $2^3$ and $3^3$, we may say that $6^3 = 2^3 \cdot 3^3 = 8 \cdot 27 = 216$.

Practice Problems

1. Evaluate $18^3$.
   Notice that $18 = 2 \cdot 9$ so this is the same as asking $2^3 \cdot 9^3$. Evaluating them separately, we get that $2^3 = 8$ and $9^3 = 729$. Multiplying them, we get $18^3 = 8 \cdot 729 = 5832$.
2. Evaluate $12^5$.
   $12 = 3 \cdot 4$ so we may convert the question into $3^5 \cdot 4^5$. We get that $3^5 = 243$ and $4^5 = 1024$. Multiplying them, we get $12^5 = 243 \cdot 1024 = 248832$.
3. Evaluate $15^2$.

Doing the same trick as the other practice problems, we may say $15^2 = 5^2 \cdot 3^2 = 25 \cdot 9 = 225$.