Radicals and complex numbers

Roots

\(\sqrt{a}\) is the nonnegative square root. \(\sqrt[3]{a}\) is the cube root; it can be negative. An even root of a negative number is not a real number. In \(\sqrt[n]{a}\), \(n\) is the index and \(a\) the radicand.

Example: \(\sqrt{49} = 7\); \(\;\sqrt[3]{-27} = -3\); \(\;\sqrt{-9}\) is not real.

Simplifying radicals

\(\sqrt{ab} = \sqrt{a}\sqrt{b}\) (for \(a, b \ge 0\)). Pull out the largest perfect-square factor (perfect cube for \(\sqrt[3]{\ }\)). For variables (assume nonnegative), divide the exponent by the index.

Example: \(\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}\); \(\;\sqrt[3]{-54} = \sqrt[3]{-27 \cdot 2} = -3\sqrt[3]{2}\).

Example: \(\sqrt{x^6} = x^3\); \(\;\sqrt{50x^5} = 5x^2\sqrt{2x}\).

Roots of fractions

\(\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}\) (for \(a \ge 0\), \(b \gt 0\)).

Example: \(\sqrt{\dfrac{9}{16}} = \dfrac{3}{4}\).

Rational exponents

Rational exponent: \(a^{1/n} = \sqrt[n]{a}\) and \(a^{m/n} = \left(\sqrt[n]{a}\right)^m\). The bottom is the root, the top is the power. All exponent rules still work.

Example: \(27^{2/3} = 3^2 = 9\); \(\;16^{-3/4} = \dfrac{1}{2^3} = \dfrac{1}{8}\); \(\;x^{1/2} \cdot x^{1/3} = x^{5/6}\).

Watch out: \(16^{-1/2} = \dfrac{1}{4}\), not \(-4\).

Adding and subtracting radicals

Only like radicals combine (same index and same radicand). Simplify each radical first.

Example: \(3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}\); \(\;\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}\).

Watch out: \(\sqrt{2} + \sqrt{3} \ne \sqrt{5}\).

Multiplying radicals

\(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\) (for \(a, b \ge 0\); for negatives, take out \(i\) first), then simplify. Multiply binomials with FOIL. Conjugates remove the radical: \((a + \sqrt{b})(a - \sqrt{b}) = a^2 - b\).

Example: \(\sqrt{6} \cdot \sqrt{10} = \sqrt{60} = 2\sqrt{15}\); \(\;(2 + \sqrt{3})(1 - \sqrt{3}) = 2 - 2\sqrt{3} + \sqrt{3} - 3 = -1 - \sqrt{3}\).

Example: \((5 + \sqrt{2})(5 - \sqrt{2}) = 25 - 2 = 23\).

Rationalizing the denominator

Rationalize: multiply top and bottom by what makes the bottom radical-free. One square root: multiply by it. A cube root: multiply to make a perfect cube. A binomial: multiply by its conjugate.

Example: \(\dfrac{6}{\sqrt{3}} = \dfrac{6\sqrt{3}}{3} = 2\sqrt{3}\); \(\;\dfrac{5}{\sqrt[3]{2}} = \dfrac{5\sqrt[3]{4}}{\sqrt[3]{8}} = \dfrac{5\sqrt[3]{4}}{2}\).

Example: \(\dfrac{4}{3 - \sqrt{5}} = \dfrac{4(3 + \sqrt{5})}{9 - 5} = 3 + \sqrt{5}\).

The imaginary unit \(i\)

The imaginary unit is \(i = \sqrt{-1}\), so \(i^2 = -1\). For \(a \gt 0\): \(\sqrt{-a} = i\sqrt{a}\). A complex number is written \(a + bi\).

Example: \(\sqrt{-18} = i\sqrt{18} = 3i\sqrt{2}\).

Watch out: take out the \(i\) first: \(\sqrt{-4} \cdot \sqrt{-9} = (2i)(3i) = 6i^2 = -6\), not \(\sqrt{36} = 6\).

Adding and multiplying complex numbers

Add real parts and imaginary parts separately. Multiply with FOIL, then replace \(i^2\) with \(-1\).

Example: \((3 + 2i) + (1 - 5i) = 4 - 3i\); \(\;(2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2 = 11 + 10i\).

Dividing complex numbers

Multiply top and bottom by the conjugate of the bottom. Use \((a + bi)(a - bi) = a^2 + b^2\).

Example: \(\dfrac{3 + i}{1 - i} = \dfrac{(3 + i)(1 + i)}{(1 - i)(1 + i)} = \dfrac{2 + 4i}{2} = 1 + 2i\).

Powers of \(i\)

They repeat every 4: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), \(i^4 = 1\). Divide the exponent by 4 and use the remainder.

Example: \(i^{23}\): \(23 = 4 \cdot 5 + 3\), so \(i^{23} = i^3 = -i\).