Operations with radicals

Simplify the following mathematical expression by extracting factors from the radicals and grouping like terms:

$5\sqrt{12} - \sqrt{27} + 2\sqrt{75}$

A technology company is designing a new smartphone model. The screen is rectangular and its exact dimensions are given by the following radicals: the base measures $\sqrt{12}$ cm and the height measures $\sqrt{3}$ cm. Calculate the area of the screen, expressing the result in simplified form.

Extract all possible factors from the following radicals to obtain the simplest expression:

1. a) $\sqrt{72x^4y^3}$

1. b) $\sqrt[3]{128a^5}$

Extract all possible factors from the following radicals and simplify the resulting expression to obtain a single term:

$3\sqrt{80} - 2\sqrt{45} + \sqrt{20}$

Rationalise the following fractions so that no radical remains in the denominator. Remember to simplify the final result if possible:

1. a) $\frac{10}{\sqrt{2}}$

1. b) $\frac{3}{\sqrt[3]{5}}$

In the design of a new smartphone chip, one of the connections has a length given by the product of two radical components. Calculate the exact value of this length by simplifying the following product and expressing it under a single radical:

$\sqrt[3]{a^2} \cdot \sqrt[4]{a^3}$

Calculate the area of a square public square whose side measures $(2 + \sqrt{5})$ metres. Express the result in exact form, simplifying any radicals that appear when expanding the binomial.

Rationalise the following expressions to eliminate the radicals from the denominator and simplify the final result:

1. a) $\dfrac{6}{\sqrt[3]{9}}$

1. b) $\dfrac{10}{\sqrt{5}}$

Simplify the following combined expression as much as possible, extracting factors from the radicals and rationalising any terms necessary to group the numerical part:

$$3\sqrt{175} - \frac{5}{\sqrt{7}} + \sqrt[4]{441} - 2\sqrt{112}$$

Rationalise and simplify the following fraction. You must work out the numerator after rationalising to obtain the simplest possible expression:

$$\frac{2\sqrt{6} - 3\sqrt{2}}{2\sqrt{3} + \sqrt{2}}$$

Reduce to a single radical and simplify the resulting expression, assuming that the variables $x$ and $y$ represent positive real values:

$$\frac{\sqrt[3]{x^2 \cdot y} \cdot \sqrt{x \cdot y^3}}{\sqrt[6]{x^5 \cdot y^4}}$$

An architect is designing a triangular skylight for a new shopping centre. The design requires the base of the skylight to be an equilateral triangle whose height is exactly $H = \sqrt{18} + \sqrt{6}$ metres. Calculate the exact value of the area of the skylight, expressing the result as a simplified radical in $m^2$.