Exercises: Properties of radicals and rationalisation

Extract all possible factors from the following radicals:

1. a) $\sqrt{72}$
2. b) $\sqrt[3]{250}$

Introduce the integer factors into the radical and simplify if possible:

1. a) $3 \sqrt{5}$
2. b) $2 \sqrt[4]{3}$

Simplify the following radicals by dividing the index and the exponent by their greatest common factor:

1. a) $\sqrt[6]{27}$
2. b) $\sqrt[8]{16}$

Perform the following multiplications of radicals with the same index and extract factors if possible:

1. a) $\sqrt{2} \cdot \sqrt{8}$
2. b) $\sqrt[3]{4} \cdot \sqrt[3]{10}$

Perform the following additions and subtractions of like radicals:

1. a) $5\sqrt{3} - 2\sqrt{3} + 4\sqrt{3}$
2. b) $\frac{1}{2}\sqrt{7} + \frac{3}{2}\sqrt{7} - \sqrt{7}$

Rationalize the denominator of the following fractions and simplify the result:

1. a) $\frac{6}{\sqrt{3}}$
2. b) $\frac{10}{\sqrt{5}}$

Rationalize the following expressions by multiplying by the conjugate of the denominator:

1. a) $\frac{4}{\sqrt{5}-1}$
2. b) $\frac{2}{\sqrt{3}+1}$

Express as a single radical and simplify the result:

1. a) $\sqrt{\sqrt[3]{64}}$
2. b) $\left(\sqrt[3]{\sqrt{2}}\right)^6$

Perform the following operation, first extracting the necessary factors so that the radicals are like radicals:

$3\sqrt{50} + 2\sqrt{18} - \sqrt{8}$

Reduce to a common index and perform the following multiplication and division of radicals:

$\frac{\sqrt{2} \cdot \sqrt[3]{3}}{\sqrt[6]{6}}$

Rationalize the following fractions with roots of index greater than 2:

1. a) $\frac{10}{\sqrt[5]{8}}$
2. b) $\frac{3}{\sqrt[4]{27}}$

Rationalize the denominator and fully simplify the following expression:

$\frac{\sqrt{2}}{\sqrt{3}+\sqrt{2}}$

In a carpentry workshop, a rectangular board is to be cut with exact dimensions of $\sqrt{12}$ in in width and $\sqrt{27}$ in in length. Calculate the area and the perimeter of the board, expressing the results exactly using simplified radicals.

The period $T$ of oscillation of a pendulum is calculated with the formula $T = 2\pi \sqrt{\frac{l}{g}}$. If we have a pendulum of length $l = 12$ feet and we consider gravity to be $g = 10$ ft/s$^2$, find the exact value of the period $T$. Rationalize the denominator and extract any possible factors.

Simplify the following expression by introducing the factors into the innermost radical and expressing it as a single root:

$\sqrt{2 \cdot \sqrt[3]{2}}$

Calculate the exact length of the main diagonal $D$ of a cube whose edge measures $a = \sqrt{5}$ in. Remember that the diagonal of a cube is calculated as $D = \sqrt{a^2 + a^2 + a^2}$. Express the result in its simplest possible form.

Perform the following combined operation with irrational fractions. To do this, first rationalize each fraction separately and then carry out the subtraction:

$\frac{2}{\sqrt{5}-\sqrt{3}} - \frac{2}{\sqrt{5}+\sqrt{3}}$

In graphic design, proportions based on the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ are considered aesthetically perfect. A canvas has the shape of a golden rectangle (the ratio of its longer side to its shorter side is $\phi$). If the shorter side of the canvas measures exactly $\sqrt{5}-1$ feet, calculate the exact length of the longer side. Fully simplify the result.

Extract all possible factors from the following root and simplify the resulting expression:

$\sqrt[3]{\frac{81 x^4 y^7}{16 z^5}}$

The facade of a cabin has the shape of an isosceles triangle. The base of the triangle measures $\sqrt{6}$ feet and the two equal sides measure $\sqrt{15}$ feet each. Calculate the exact value of the height $h$ above the base and the total area of the facade. Rationalize the results if roots appear in the denominator.

Simplify the following expression with nested radicals, expressing the result as a single root:

$\sqrt{2 \cdot \sqrt[3]{4 \cdot \sqrt{8}}}$

Rationalize the denominator of the following fraction and fully simplify by operating on the numerator:

$\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}+\sqrt{2}}$

In physics, the speed $v$ of an object can be found from its kinetic energy $E_c$ and its mass $m$ using the formula $v = \sqrt{\frac{2E_c}{m}}$. If a prototype vehicle has a mass of $m = \sqrt{3}$ lb and its kinetic energy is $E_c = 45$ J, calculate the exact value of its speed. Rationalize and extract any possible factors from the resulting root.

Prove, by expanding the special products and simplifying, that the result of the following expression is an integer:

$(\sqrt{7} - \sqrt{3})^2 + (\sqrt{7} + \sqrt{3})^2$