Properties of radicals and rationalising the denominator

Take out as many factors as possible from the following radicals:

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

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

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

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 additions and subtractions of like radicals:

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

Perform the following operation, first taking out the necessary factors so that the radicals are like terms:

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

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

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

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

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

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

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

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

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

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

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

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

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

Rationalise the denominator of the following fraction and simplify as much as possible by working out the numerator:

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

In a carpentry workshop, a rectangular board with exact dimensions of $\sqrt{12}$ dm wide and $\sqrt{27}$ dm long is going to be cut. Calculate the area and perimetre of the board, expressing the results in exact form using simplified radicals.

Calculate the exact length of the main diagonal $D$ of a cube with an edge length of $a = \sqrt{5}$ cm. Remember that the diagonal of a cube is calculated as $D = \sqrt{a^2 + a^2 + a^2}$. Express the result in the most simplified form possible.

The front wall of a cabin is in the shape of an isosceles triangle. The base of the triangle measures $\sqrt{6}$ metres and the two equal sides measure $\sqrt{15}$ metres each. Calculate the exact value of the height $h$ perpendicular to the base and the total area of the front wall. Rationalise the results if roots appear in the denominator.

In physics, the speed $v$ of an object can be rearranged 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 $m = \sqrt{3}$ kg and its kinetic energy is $E_c = 45$ J, calculate the exact value of its speed. Rationalise and take out any possible factors from the resulting root.