Radicals and rational exponents
Roots undo powers. Writing roots as fractional exponents lets the same exponent rules simplify radicals, which appear throughout geometry, physics, and statistics.
Square roots
The square root of a non-negative number a is the non-negative number whose square is a: \(\sqrt{49} = 7\). Both 7 and -7 square to 49, but the symbol \(\sqrt{\ }\) means the principal, non-negative root. Square roots of negative numbers are not real numbers; they lead to complex numbers, met later.
Higher roots
The nth root \(\sqrt[n]{a}\) is the number whose nth power is a: \(\sqrt[3]{-8} = -2\) because \((-2)^3 = -8\), and \(\sqrt[4]{81} = 3\). Odd roots of negative numbers are real; even roots of negative numbers are not.
Simplifying square roots
Use \(\sqrt{ab} = \sqrt{a}\sqrt{b}\) to pull out perfect-square factors: \(\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}\). A square root is simplified when no perfect-square factor other than 1 remains under it. Note that \(\sqrt{a + b}\) is not \(\sqrt{a} + \sqrt{b}\): \(\sqrt{9 + 16} = 5\), not 7.
Adding and multiplying radicals
Like radicals, with the same index and radicand, combine like like terms: \(3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}\). Sometimes simplifying first reveals like radicals: \(\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}\). Multiply radicals by multiplying radicands: \(\sqrt{6}\sqrt{10} = \sqrt{60} = 2\sqrt{15}\).
Rationalising denominators
By convention, simplified answers avoid radicals in the denominator. Multiply top and bottom by the radical: \(\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}\). For a denominator like \(1 + \sqrt{2}\), multiply by the conjugate \(1 - \sqrt{2}\), using \((a + b)(a - b) = a^2 - b^2\).
Rational exponents
Roots can be written as fractional exponents: \(a^{1/n} = \sqrt[n]{a}\), and \(a^{m/n} = (\sqrt[n]{a})^m\). So \(27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9\), and \(16^{-1/2} = \frac{1}{4}\). Take the root first to keep numbers small.
Using exponent rules on roots
Because roots are exponents, all the exponent rules apply: \(\sqrt{x} \cdot \sqrt[3]{x} = x^{1/2} \cdot x^{1/3} = x^{5/6}\). This turns awkward radical expressions into simple exponent arithmetic, a technique used constantly in calculus.
Worked example
The diagonal of a square with side 6 cm has length \(\sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2} \approx 8.49\) cm. Leaving the exact answer \(6\sqrt{2}\) shows the pattern: every square's diagonal is its side times \(\sqrt{2}\).
Key takeaways
The square root symbol means the non-negative root; even roots of negatives are not real, while odd roots of negatives are. Simplify square roots by pulling out perfect-square factors, and remember that the root of a sum is not the sum of roots. Combine only like radicals, simplifying first when needed. Rationalise denominators, using the conjugate for binomial denominators. Rational exponents express roots: a to the m/n is the nth root of a, raised to the m. All exponent rules then apply to radicals.
Flashcard
The non-negative root
Check your understanding
Simplify \(\sqrt{72}\).