Lesson 1: Raising Equations to Powers

Property

The Product of Conjugates formula is $(a - b)(a + b) = a^2 - b^2$.
When expressions like $(x - \sqrt{y})$ and $(x + \sqrt{y})$ are multiplied, the result contains no square roots.

Examples

• To simplify $(6 - \sqrt{5})(6 + \sqrt{5})$, use the pattern $a^2 - b^2$: $6^2 - (\sqrt{5})^2 = 36 - 5 = 31$.
• For $(3 - 4\sqrt{2})(3 + 4\sqrt{2})$, the result is $3^2 - (4\sqrt{2})^2$. This simplifies to $9 - (16 \cdot 2) = 9 - 32 = -23$.
• For $(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3})$, the result is $(\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4$.

Explanation

Multiplying conjugates, which have the same terms but opposite signs, is a special trick. The middle terms with radicals cancel out, leaving a rational number. This is because the 'Outer' and 'Inner' products are opposites.

Property

When solving complex power equation problems, systematically catalog all given information and identify which algebraic identities can connect the given data to the desired result: $(a + b)^2 = a^2 + 2ab + b^2$, $(a - b)^2 = a^2 - 2ab + b^2$, $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$, and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Examples

Property

When raising equations to powers creates multiple equations with the same variables, solve the resulting system by substitution or elimination: If $u + v = a$ and $u^2 + v^2 = b$, then $(u + v)^2 = u^2 + 2uv + v^2 = a^2$, so $2uv = a^2 - b$.

Examples