To calculate [tex]\(\frac{27b^2}{a^2}\)[/tex] given that [tex]\((a + 3b)(a - 3b) = 0\)[/tex], we can follow these steps:
1. Understand the given expression:
The expression given is [tex]\((a + 3b)(a - 3b) = 0\)[/tex].
2. Identify the pattern:
This expression is a difference of squares. The formula for the difference of squares is:
[tex]\[
(x + y)(x - y) = x^2 - y^2
\][/tex]
By comparing, we have [tex]\(x = a\)[/tex] and [tex]\(y = 3b\)[/tex]. So:
[tex]\[
(a + 3b)(a - 3b) = a^2 - (3b)^2
\][/tex]
3. Simplify the expression:
[tex]\[
a^2 - (3b)^2 = a^2 - 9b^2
\][/tex]
Given that [tex]\((a + 3b)(a - 3b) = 0\)[/tex], we have:
[tex]\[
a^2 - 9b^2 = 0
\][/tex]
4. Solve for [tex]\(a^2\)[/tex]:
To isolate [tex]\(a^2\)[/tex], add [tex]\(9b^2\)[/tex] to both sides:
[tex]\[
a^2 = 9b^2
\][/tex]
5. Calculate [tex]\(\frac{27b^2}{a^2}\)[/tex]:
- Substitute [tex]\(a^2 = 9b^2\)[/tex] into the expression [tex]\(\frac{27b^2}{a^2}\)[/tex]:
[tex]\[
\frac{27b^2}{a^2} = \frac{27b^2}{9b^2}
\][/tex]
- Simplify the fraction:
[tex]\[
\frac{27b^2}{9b^2} = \frac{27}{9} = 3
\][/tex]
Thus, the value of [tex]\(\frac{27b^2}{a^2}\)[/tex] is:
[tex]\[
\boxed{27}
\][/tex]
