To multiply and simplify the expression [tex]\((6 + 6t)^2\)[/tex], we can use the binomial theorem or simply expand it using the distributive property (also known as the FOIL method). Let's go through it step-by-step:
### Binomial Theorem Approach:
The binomial theorem states:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
In this case, [tex]\(a = 6\)[/tex] and [tex]\(b = 6t\)[/tex].
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
6^2 = 36
\][/tex]
2. Calculate [tex]\(2ab\)[/tex]:
[tex]\[
2 \cdot 6 \cdot 6t = 72t
\][/tex]
3. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
(6t)^2 = 36t^2
\][/tex]
### Putting it all together:
The expanded form is:
[tex]\[
6^2 + 2 \cdot 6 \cdot 6t + (6t)^2 = 36 + 72t + 36t^2
\][/tex]
Therefore,
[tex]\[
(6 + 6t)^2 = 36 + 72t + 36t^2
\][/tex]
### Final Answer:
[tex]\[
\boxed{36 + 72t + 36t^2}
\][/tex]
