Properties of Radicals and Rationalization

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

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

Insert 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 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) $\left(\sqrt[3]{\sqrt{2}}\right)^6$

Take out as many factors as possible from the following root and simplify the resulting expression:

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

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:

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

Show, by expanding the algebraic identities and simplifying, that the result of the following expression is an integer:

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

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

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

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

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

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

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

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

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

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 and simplify the following expression as much as possible:

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

Perform the following combined operation with irrational fractions. To do so, first rationalise each fraction separately and then perform the subtraction:

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

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.

The speed $v$ (in m/s) of a wave travelling along a stretched string is calculated using the formula $v = \sqrt{\frac{F}{\mu}}$, where $F$ is the tension in newtons and $\mu$ the linear density in kg/m. If a guitar string is subjected to a tension of $F = 12$ N and has a density of $\mu = 5$ kg/m, find the exact value of $v$. Rationalise the denominator and take out any possible factors.

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.

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 quotient of its longer side and its shorter side is $\phi$). If the shorter side of the canvas measures exactly $\sqrt{5}-1$ meters, calculate the exact length of the longer side. Simplify the result as much as possible.

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