Exercises: Properties of radicals and rationalisation

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) $\sqrt[8]{16}$

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) $\sqrt[3]{4} \times \sqrt[3]{10}$

Perform the following additions and subtractions of like radicals:

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

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) $\frac{2}{\sqrt{3}+1}$

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 taking out the necessary factors so that the radicals are like terms:

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

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

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

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:

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

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.

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

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

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.

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

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

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.

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

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

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.

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 fraction and simplify as much as possible by working out the numerator:

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

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.

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

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