Let's expand and simplify the given expression [tex]\((x^3 + 9)^2\)[/tex].
Step 1: Start with the given expression
[tex]\[
(x^3 + 9)^2
\][/tex]
Step 2: Recognize that we need to apply the binomial theorem or the formula for the square of a binomial. The square of a binomial [tex]\((a + b)^2\)[/tex] can be expanded using the formula:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
Step 3: Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in our context. Here, [tex]\(a = x^3\)[/tex] and [tex]\(b = 9\)[/tex].
Step 4: Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the binomial square expansion formula:
[tex]\[
(x^3 + 9)^2 = (x^3)^2 + 2 \cdot (x^3) \cdot 9 + 9^2
\][/tex]
Step 5: Simplify each term individually:
- [tex]\((x^3)^2 = x^6\)[/tex]
- [tex]\(2 \cdot (x^3) \cdot 9 = 18x^3 \)[/tex]
- [tex]\(9^2 = 81\)[/tex]
Step 6: Combine all the simplified terms:
[tex]\[
(x^3 + 9)^2 = x^6 + 18x^3 + 81
\][/tex]
Thus, the expanded form of [tex]\((x^3 + 9)^2\)[/tex] is:
[tex]\[
x^6 + 18x^3 + 81
\][/tex]
This is the final expanded and simplified expression.
