Expanding expressions with surds

Expand brackets containing surds, use the special expansions with surds, and meet conjugates, which multiply to give a rational number.

Expanding with surds

Surds expand exactly like pronumerals. Treat $\sqrt{3}$ like $x$, and remember that $\sqrt{3} \times \sqrt{3} = 3$.

Example 1●

Expand $\sqrt{2}(5 - 3\sqrt{2})$.

$= 5\sqrt{2} - 3 \times 2$

$= 5\sqrt{2} - 6$

Example 2●●

Expand $(2 + \sqrt{5})(3 - \sqrt{5})$.

$= 6 - 2\sqrt{5} + 3\sqrt{5} - 5$

Four products, just like (a + b)(c + d).

$= 1 + \sqrt{5}$

Example 3: perfect square●●

Expand $(\sqrt{7} - 2)^2$.

$= 7 - 4\sqrt{7} + 4$

A² − 2AB + B² with A = √7 and B = 2.

$= 11 - 4\sqrt{7}$

Example 4●●

Expand $(3\sqrt{2} + \sqrt{5})(3\sqrt{2} - \sqrt{5})$.

$= 18 - 5$

Difference of squares: (3√2)² − (√5)².

$= 13$

Flashcards

√a × √a

$a$

(a + √b)(a − √b)

$a^2 - b$

(k√b)²

$k^2 b$

Practice

$(1 + \sqrt{3})^2 =$
Evaluate $(\sqrt{11} + 3)(\sqrt{11} - 3)$.
The conjugate of $5 - 2\sqrt{3}$ is:
$\sqrt{3}(\sqrt{3} + 2) =$
$(\sqrt{5} + 1)(\sqrt{5} + 3) =$
Evaluate $(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})$.
$(2\sqrt{3} + 1)^2 =$
$(3 - \sqrt{2})^2 = a - b\sqrt{2}$. Find $a$.