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The expression you need to factorize is:

\[
\frac{a^2}{9} - 49b^2
\]

This expression can be recognized as a difference of squares, which has the general form:

\[
x^2 - y^2 = (x - y)(x + y)
\]

Here, we can rewrite the given expression as:

\[
\left(\frac{a}{3}\right)^2 - (7b)^2
\]

Now, applying the difference of squares formula:

\[
\frac{a^2}{9} - 49b^2 = \left(\frac{a}{3} - 7b\right)\left(\frac{a}{3} + 7b\right)
\]

Thus, the completely factorized form of the expression is:

\[
\boxed{\left(\frac{a}{3} - 7b\right)\left(\frac{a}{3} + 7b\right)}
\]

Would you like further details or have any questions?

Here are some related questions:

1. What are some examples of factoring other types of polynomials?
2. How do you factor a trinomial completely?
3. Can you factorize expressions that are not a perfect square?
4. What is the significance of recognizing patterns like the difference of squares in algebra?
5. How do you apply the difference of squares in solving quadratic equations?

**Tip:** Recognizing common patterns such as the difference of squares can make factoring polynomials much quicker and easier.

Factorise completely \( \frac{a^2}{9} - 49b^2 \)

Learn how to factorize the algebraic expression \( \frac{a^2}{9} - 49b^2 \) completely using the difference of squares method. This problem explores key algebraic concepts and is suitable for students in Grades 9-11.

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