Real Numbers: Radicals and Rationalizing Denominators

Simplifying radicals and rationalizing

For $a, b > 0$, $\sqrt{a}\sqrt{b} = \sqrt{ab}$ and $\sqrt{k^2 a} = k\sqrt{a}$. Simplify each radical first, then combine like radicals.

Example: $\sqrt{12} + \sqrt{27} - \sqrt{48} = 2\sqrt{3} + 3\sqrt{3} - 4\sqrt{3} = \sqrt{3}$

When the denominator is a sum or difference, use $(a + b)(a - b) = a^2 - b^2$ to remove the radical. Final answers should have no radical in the denominator.

Nested radicals and absolute values

Look for two numbers with sum $a + b$ and product $ab$. If there is no factor 2, as in $\sqrt{4 + \sqrt{15}}$, rewrite the inside as $\frac{8 + 2\sqrt{15}}{2}$ first.

Absolute value: $|a| = a$ if $a \geqq 0$ and $|a| = -a$ if $a < 0$. Also $\sqrt{a^2} = |a|$, so $\sqrt{(x - 1)^2}$ equals $x - 1$ or $1 - x$ depending on $x$.

Integer and fractional parts; symmetric values

Since $2 < \sqrt{7} < 3$, the integer part of $\sqrt{7}$ is 2 and the fractional part is $\sqrt{7} - 2$ (the number minus its integer part).

If $x + \frac{1}{x} = k$, then