Roots and radicals

Square roots, simplifying, rationalizing, nested roots, n-th roots and rational exponents. 8 problems with step-by-step solutions.

Solve each problem in your notebook first, then open the solution.

Problem 1 Square roots

a) Evaluate exactly or to two decimals: \(\sqrt{729}\), \(\sqrt{1.44}\), \(\sqrt{0.0049}\), \(\sqrt{20}\).

b) Solve \(x^2 = 49\) and \(\sqrt{x} = 7\).

Show solution

a) \(27\); \(1.2\); \(0.07\); \(\approx 4.47\).

b) \(x = \pm 7\); \(x = 49\).

Problem 2 Simplifying with conditions

a) \(0 < a < 3\). Simplify \(\sqrt{a^2 - 6a + 9}\).

b) \(x < 0 < y\). Simplify \(\sqrt{x^2} + \sqrt{(x - y)^2} + \sqrt{y^2}\).

c) \(-2 < x < 0\). Simplify \(\sqrt{(x + 2)^2} + \sqrt{x^2}\).

Show solution

\(\sqrt{t^2} = |t|\).

a) \(|a - 3| = 3 - a\).

b) \(-x + (y - x) + y = 2y - 2x\).

c) \((x + 2) + (-x) = 2\).

Problem 3 Simplifying roots

a) Simplify \(\sqrt{72}\), \(\sqrt{300}\), \(\sqrt{98}\), \(\sqrt{0.08}\).

b) Write as \(\sqrt{x}\): \(3\sqrt5\), \(4\sqrt{3}\), \(\frac12\sqrt{20}\).

c) Calculate \(\sqrt{50} - \sqrt{18} + \sqrt{32} - \sqrt{8}\).

Show solution

a) \(6\sqrt2\); \(10\sqrt3\); \(7\sqrt2\); \(\sqrt{\frac{8}{100}} = \frac{2\sqrt2}{10} = \frac{\sqrt2}{5}\).

b) \(\sqrt{45}\); \(\sqrt{48}\); \(\sqrt5\).

c) \(5\sqrt2 - 3\sqrt2 + 4\sqrt2 - 2\sqrt2 = 4\sqrt2\).

Problem 4 Operations with roots

a) Calculate \((3\sqrt2 - \sqrt3)(\sqrt2 + 2\sqrt3)\).

b) Calculate \((\sqrt7 + \sqrt3)(\sqrt7 - \sqrt3)\) and \((\sqrt5 - 2)^2\).

c) Compare \(\sqrt7 + \sqrt2\) and \(\sqrt6 + \sqrt3\).

Show solution

a) \(6 + 6\sqrt6 - \sqrt6 - 6 = 5\sqrt6\).

b) \(7 - 3 = 4\); \(5 - 4\sqrt5 + 4 = 9 - 4\sqrt5\).

c) Square both: \(9 + 2\sqrt{14}\) and \(9 + 2\sqrt{18}\). Since \(14 < 18\), \(\sqrt7 + \sqrt2 < \sqrt6 + \sqrt3\).

Problem 5 Rationalizing

Rationalize the denominator.

a) \(\dfrac{6}{\sqrt3}\) b) \(\dfrac{5}{2\sqrt{10}}\) c) \(\dfrac{4}{\sqrt5 - 1}\) d) \(\dfrac{\sqrt3 + \sqrt2}{\sqrt3 - \sqrt2}\) e) \(\dfrac{1}{\sqrt2 + \sqrt3 + \sqrt5}\)

Show solution

a) \(\frac{6\sqrt3}{3} = 2\sqrt3\)
b) \(\frac{5\sqrt{10}}{20} = \frac{\sqrt{10}}{4}\)
c) \(\frac{4(\sqrt5 + 1)}{4} = \sqrt5 + 1\)
d) \(\frac{(\sqrt3 + \sqrt2)^2}{1} = 5 + 2\sqrt6\)
e) Multiply by \(\sqrt2 + \sqrt3 - \sqrt5\): the denominator becomes \((\sqrt2 + \sqrt3)^2 - 5 = 2\sqrt6\). Then \(\frac{(\sqrt2 + \sqrt3 - \sqrt5)\sqrt6}{12} = \frac{2\sqrt3 + 3\sqrt2 - \sqrt{30}}{12}\)

Problem 6 Nested roots

Simplify.

a) \(\sqrt{7 - 4\sqrt3}\) b) \(\sqrt{11 + 2\sqrt{30}}\) c) \(\sqrt{5 - 2\sqrt6} + \sqrt{5 + 2\sqrt6}\) d) \(\sqrt{12 + \sqrt{12 + \sqrt{12 + \cdots}}}\)

Show solution

Use \(\sqrt{a + b \pm 2\sqrt{ab}} = \sqrt{a} \pm \sqrt{b}\) (with \(a > b\)).

a) \(7 - 2\sqrt{12} = (\sqrt4 - \sqrt3)^2\): \(2 - \sqrt3\)
b) \(\sqrt6 + \sqrt5\)
c) \((\sqrt3 - \sqrt2) + (\sqrt3 + \sqrt2) = 2\sqrt3\)
d) \(x = \sqrt{12 + x}\), \(x^2 - x - 12 = 0\), \(x = 4\) (positive root)

Problem 7 n-th roots

a) Evaluate \(\sqrt[3]{-125}\), \(\sqrt[4]{81}\), \(\sqrt[5]{0.00032}\).

b) Calculate \(\sqrt[3]{64} - \sqrt[4]{16} + \sqrt{0.25} - \sqrt[3]{0.008}\).

c) Simplify \(\sqrt[4]{16x^8y^4}\) for \(y < 0\), and \(\sqrt[3]{-27a^6}\).

Show solution

a) \(-5\); \(3\); \(0.2\).

b) \(4 - 2 + 0.5 - 0.2 = 2.3\).

c) \(2x^2|y| = -2x^2y\); \(-3a^2\).

Problem 8 Rational exponents

a) Evaluate \(27^{2/3}\), \(16^{-3/4}\), \(32^{0.4}\).

b) Simplify (all variables positive): \((4a^{1/3})(2a^{2/3})\), \((x^{3/4})^{8}\cdot x^{-5}\), \(\sqrt{a}\cdot\sqrt[3]{a}\cdot\sqrt[6]{a}\).

Show solution

a) \((\sqrt[3]{27})^2 = 9\); \(\frac{1}{(\sqrt[4]{16})^3} = \frac18\); \(32^{2/5} = 2^2 = 4\).

b) \(8a\); \(x^6\cdot x^{-5} = x\); \(a^{1/2 + 1/3 + 1/6} = a\).