Operations with Radicals

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

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

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

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

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

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

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

Insert the factors inside the radical in the following expressions:

1. a) $3 \sqrt{5x}$

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

Find the height ($h$) of a triangular solar panel if we know that its area is $\sqrt{24}$ square meters and its base measures $\sqrt{2}$ meters. Remember that the area of a triangle is calculated as $A = \frac{\text{base} \times \text{altura}}{2}$. Express the result simplifying the radicals.

Perform the following operations with radicals of the same index and express the result in the simplest possible form:

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

1. b) $\frac{\sqrt[3]{81}}{\sqrt[3]{3}}$

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:

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

In a graphic design studio, work is being done on a logo that uses a spiral based on rectangles. One of the critical segments has a length $L$ defined by the operation between the following radicals with different indices. Calculate its exact value and simplify it:

$$L = \frac{\sqrt[4]{32} \times \sqrt[3]{16}}{\sqrt[6]{2}}$$

Apply the properties of radicals to reduce to a single root and simplify the final result:

$\sqrt[3]{\sqrt{64x^{12}}}$

Reduce the following expression to a single radical as simple as possible, applying the properties of a root of a root and the extraction of factors:

$\sqrt{\sqrt[3]{64x^8}}$

Simplify the following expression with nested radicals to convert it into a power with a fractional exponent with base $a$:

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

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.

Perform the following combined operation with radicals, applying the distributive property and simplifying like terms:

$(\sqrt{3} + 2) \times (2\sqrt{3} - 5)$

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

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) $\frac{3}{\sqrt[3]{5}}$

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) $\dfrac{10}{\sqrt{5}}$

An interior decorator wants to place a wooden moulding around a large square mirror that has an area of $20 \text{ m}^2$. Express the perimeter of the mirror in simplified radical form (extract factors if necessary).

An architect needs to calculate the pressure exerted by a structure on the ground. After completing the initial calculations, the resulting expression is as follows. Rationalise it using the conjugate of the denominator and simplify it as much as possible:

$\dfrac{4}{\sqrt{7} - \sqrt{3}}$

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

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

Rationalise the following expression. Note that, as there are three square roots in the denominator, you may need to multiply by the conjugate more than once:

$$\frac{4}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$$

Rationalise the following expression and simplify the result by extracting all possible factors from the radicals:

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

Is the result a rational or irrational number? Justify your answer by analysing the structure of the number obtained.

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

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

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

A metal structure in the shape of a rectangular prism is to be built, whose dimensions (width, length and height) are given by the following expressions in meters:

• $a = \sqrt{2} + \sqrt{3}$

• $b = \sqrt{2} - \sqrt{3} + \sqrt{6}$

• $c = \sqrt{6}$

Calculate the total volume of the structure. The result must be a real number expressed in exact form (with radicals if necessary).