To expand the given expression [tex]\(\left(a + \frac{1}{3a}\right)^2\)[/tex], we can follow these specific steps in algebra:
1. Understand the expression: We start with [tex]\(\left(a + \frac{1}{3a}\right)^2\)[/tex]. This indicates that we need to square the binomial expression [tex]\(a + \frac{1}{3a}\)[/tex].
2. Apply the binomial expansion formula: Recall that [tex]\((x + y)^2 = x^2 + 2xy + y^2\)[/tex]. In our case, [tex]\(x = a\)[/tex] and [tex]\(y = \frac{1}{3a}\)[/tex].
3. Square the first term: We square the first term [tex]\(a\)[/tex]:
[tex]\[
a^2
\][/tex]
4. Square the second term: We square the second term [tex]\(\frac{1}{3a}\)[/tex]:
[tex]\[
\left(\frac{1}{3a}\right)^2 = \frac{1^2}{(3a)^2} = \frac{1}{9a^2}
\][/tex]
5. Multiply the two terms and double the product: We find the product of [tex]\(a\)[/tex] and [tex]\(\frac{1}{3a}\)[/tex], and then double it:
[tex]\[
2 \cdot a \cdot \frac{1}{3a} = 2 \cdot \frac{1}{3} = \frac{2}{3}
\][/tex]
6. Combine all parts together: Summing these results, we get:
[tex]\[
a^2 + \frac{2}{3} + \frac{1}{9a^2}
\][/tex]
Hence, the expanded form of [tex]\(\left(a + \frac{1}{3a}\right)^2\)[/tex] is:
[tex]\[
a^2 + \frac{2}{3} + \frac{1}{9a^2}
\][/tex]
