Multiply (3$\sqrt{a}$ - 4a) by (4a + 3$\sqrt{a}$)

Analyze the question: We are multiplying (3√a − 4a)(4a + 3√a). The two brackets contain the same terms, 3√a and 4a, but with different signs. This is a difference of two squares.

Concept: (x − y)(x + y) = x² − y²

Rearrange the second bracket: (4a + 3√a) = (3√a + 4a).

So the product is (3√a − 4a)(3√a + 4a), with x = 3√a and y = 4a.

Solution:

x² = (3√a)² = 9a

y² = (4a)² = 16a²

Product = 9a − 16a²

Check by full expansion:

• 3√a × 4a = 12a√a
• 3√a × 3√a = 9a
• −4a × 4a = −16a²
• −4a × 3√a = −12a√a

The 12a√a terms cancel, leaving 9a − 16a². ✓

Tip: When you see two brackets with the same terms but opposite signs between them, think "difference of two squares" right away. Square the first term, square the second, and subtract. Also remember that (√a)² = a.