Exercises: Operations with irrational numbers

Take out all possible factors from the following radicals to express them in their simplest form:

1. a) $\sqrt{252}$
2. b) $\sqrt[3]{1080}$

Carry out the following calculation by collecting like radicals and express the final result:

$7\sqrt{5} - 2\sqrt{5} + 4\sqrt{5} - \sqrt{5}$

Multiply the following radicals and simplify the result by taking out any possible factors:

$\sqrt{15} \cdot \sqrt{10}$

Divide the following radicals of the same index and simplify the result:

$\frac{\sqrt[3]{320}}{\sqrt[3]{5}}$

Calculate the power of the following radical and express the result in its simplest form, taking out factors if necessary:

$(\sqrt[4]{27})^2$

Express as a single radical using the laws of roots and simplify the result:

$\sqrt{\sqrt[3]{64}}$

Rationalise the denominator of the following fraction and simplify the resulting expression:

$\frac{18}{\sqrt{3}}$

Rationalise the following expression to eliminate the root from the denominator:

$\frac{10}{\sqrt[4]{8}}$

Marcos is calculating the length of electrical cable needed to connect two junction boxes in a house in Cuenca. The two continuous linear sections measure $\sqrt{75}$ m and $\sqrt{12}$ m. In his notebook, he writes the following:

“$\sqrt{75} + \sqrt{12} = \sqrt{75 + 12} = \sqrt{87}\text{ m} \approx 9,33\text{ m}$”.

1. a) Explain in your own words the conceptual mistake Marcos made. Show that the square root of a sum is not the sum of the square roots by providing a simple numerical counterexample using perfect squares (such as $9$ and $16$).
2. b) Solve the operation correctly: break down the radicands into prime factors, extract all possible factors outside the radicals, and combine like terms to give the exact length in the form $k\sqrt{3}$.
3. c) Knowing that $\sqrt{3} \approx 1,732$, calculate the actual decimal value of the total length and determine how many metres of cable Marcos would have been missing if he had bought the material relying on his erroneous calculation.

The sustainability department of a town council in Castilla y León has demarcated two adjacent square plots to be used for school urban gardens. The first plot has an area of $45\text{ m}^2$ and the second, attached to the previous one, has an area of $20\text{ m}^2$. A straight fence needs to be ordered to cover the entire front of both plots (that is, a total length of $L = \sqrt{45} + \sqrt{20}$ metres).

1. a) Estimate: without using the square root key on the calculator, determine between which two consecutive integers $\sqrt{45}$ lies and between which two $\sqrt{20}$ lies. From these two bounds, indicate between which two integers the total length $L$ must safely fall.
2. b) Calculate: break down the radicands, extract factors, and write the exact length of the fence as a single simplified radical of the form $a\sqrt{5}$.
3. c) Check: using the approximation $\sqrt{5} \approx 2,24$, find the decimal value of the exact result obtained in part b and verify with reasoning whether your initial estimate in part a was correct.

A gardener needs to fence a triangular flowerbed in a botanical garden. The lengths of the three sides of the flowerbed are $\sqrt{50}$ m, $\sqrt{18}$ m and $\sqrt{98}$ m. Express the total perimeter of the flowerbed as a single simplified radical.

In a crystallography experiment, the volume of a crystal is given by the expression $V = \sqrt[3]{4} \cdot \sqrt{8}$ mm$^3$. Express with a common index, calculate and take out factors to write the exact volume in its simplest radical form.

The intensity of an electromagnetic signal at a certain point in space has been modelled using the expression $I = \frac{12}{\sqrt{5} - \sqrt{2}}$ W/m$^2$. Rationalise the expression to calculate the exact value of the intensity without roots in the denominator.

Carry out the following combined operation with radicals by applying the distributive property, and simplify the result as much as possible:

$\sqrt{3} \cdot (\sqrt{27} - \sqrt{12}) + 5\sqrt{9}$

Simplify the following algebraic expression by inserting factors under the root and reducing to a single radical, assuming that $x$ and $y$ are positive real numbers:

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

The golden ratio $\Phi = \frac{1+\sqrt{5}}{2}$ frequently appears in the proportions of art and design. If the length of a credit card is $8\Phi$ cm and its width is $8$ cm, calculate the exact area and exact perimeter of the card, expressing the results in simplified radical form.

In the structural design of a suspension bridge, the tension in one of the main cables is calculated using the formula $T = \frac{15\sqrt{2}}{3 + \sqrt{2}}$ kN. Rationalise the denominator and simplify the expression for the tension for the engineers.

The period $T$ of oscillation of a pendulum (in seconds) is given by the formula $T = 2\pi \sqrt{\frac{L}{g}}$. If the length of the pendulum is $L = \frac{9}{10}$ metres and we take gravity as $g = 10$ m/s$^2$, calculate the exact value of $T$ by operating with the irrational numbers and rationalising the result.

For the assembly of solar panel supports in an industrial warehouse in Córdoba, the length of a reinforcing aluminium profile is determined by the expression:

Two installers from the technical team debate the most appropriate procedure to calculate the exact value of $L$ without using a calculator:

• Rocío's method: first extract factors from the radicals in the numerator ($\sqrt{72}$ and $\sqrt{8}$), subtract the resulting like radicals, and finally simplify the fraction by dividing by $\sqrt{2}$.

• Carlos's method: rationalise the expression directly by multiplying the numerator and denominator by $\sqrt{2}$, apply the distributive property in the numerator, and simplify the resulting fraction.

1. a) Carry out the calculations step by step following the method proposed by Rocío.
2. b) Carry out the calculations step by step following the method proposed by Carlos.
3. c) Compare both approaches: do the numerical results match? Justify which of the two methods you consider more efficient and with less risk of error when handling large numbers.

In an aluminium joinery factory in Zaragoza, an operator programs a cutting machine according to a formula involving a positive integer constant $K$:

When requesting the machine to fully rationalise and simplify the expression by multiplying by the conjugate of the denominator, the control screen displays the exact result:

1. a) Reconstruct the inverse algebraic process step by step to find out what integer value the constant $K$ originally had.
2. b) If in a new order the final dimension is required to be exactly three times larger, that is, $15 \cdot (\sqrt{7} + \sqrt{3})\text{ mm}$, deduce with reasoning what value should be assigned to $K$ without needing to repeat the entire rationalisation from scratch. Is there a direct proportional relationship between the constant $K$ and the final dimension value? Justify your answer.

Show that the following expression represents an integer, carrying out the fraction additions and necessary rationalisations step by step:

$\frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} + \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}$

Calculate the exact area of the shaded region bounded by a circle of radius $R = 2\sqrt{6}$ cm and a regular hexagon inscribed in it. Express the result by factorising the irrational numbers as much as possible and taking out a common factor if possible.

Rationalise and simplify the following combined expression as much as possible, paying attention to signs and taking out factors:

$\frac{2\sqrt{5} + 3\sqrt{2}}{2\sqrt{5} - 3\sqrt{2}} - 6\sqrt{10}$

Solve the following linear equation by isolating the unknown $x$ and rationalizing the final result:

$x\sqrt{2} - 3 = x\sqrt{3} + \sqrt{6}$

Simplify the nested radical $\sqrt{14 - 6\sqrt{5}}$ given that it is equivalent to an expression of the form $a - b\sqrt{5}$, where $a$ and $b$ are positive integers. Find the values of $a$ and $b$ by squaring both expressions.

In a regular tetrahedron, the edge measures $a = 3\sqrt{2}$ cm. Given that the height of a regular tetrahedron in terms of its edge is $H = a\frac{\sqrt{6}}{3}$ and its volume is $V = \frac{a^3\sqrt{2}}{12}$, calculate the exact value of the height $H$ and the volume $V$, expressing the results in their simplest radical form.

In the theory of special relativity, the Lorentz factor is given by the expression $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$. Calculate the exact and rationalised value of $\gamma$ for a spacecraft travelling at a speed of $v = \frac{\sqrt{5}}{3}c$.

The diagonal of a rectangle with sides $1$ and $\sqrt{2}$ is denoted by $d_1$. If we construct a new rectangle whose sides are $d_1$ and $\sqrt{3}$, its diagonal is $d_2$. If we repeat this process so that the legs of the $n$th right-angled triangle are $d_{n-1}$ and $\sqrt{n+1}$, determine the exact value of the diagonal $d_4$ by operating step by step with the irrational numbers.

An agricultural cooperative in Albacete installs a photovoltaic irrigation system whose two main circuits have electrical resistances given by:

1. a) To simplify $R_1$, a technician hesitates between multiplying directly by the conjugate $(\sqrt{48} + \sqrt{12})$ or simplifying the denominator beforehand. Show that by extracting factors from $\sqrt{48}$ and $\sqrt{12}$ before operating, a direct subtraction of like radicals is obtained, and finish rationalising $R_1$ using the shortest possible path.
2. b) Extract factors from $R_2$ to express it in the form $k\sqrt{3}$.
3. c) Calculate the total equivalent resistance of the set connected in series, $R_T = R_1 + R_2$, combining like terms into a single simplified radical.
4. d) The technical safety specification of the installation requires the total resistance to be strictly less than $14\ \Omega$. Knowing that $1,73 < \sqrt{3} < 1,74$, bound the value of $R_T$ and justify with reasoning (without a calculator) whether the installation complies with safety regulations or if it must be rejected.

During environmental surveillance and fire prevention work in the Doñana National Park (Andalusia), a drone patrol calibrates its thermal camera. The optical distortion index $D$ is modelled by the subtraction of two conjugate factors:

1. a) Show that the exact value of $D$ reduces to a simple irrational expression of the form $a\sqrt{b}$. Work out the calculation using two different strategies: first by rationalising each fraction separately with its respective conjugate, and then by finding the common denominator of the subtraction directly using the identity $(x-y)(x+y) = x^2 - y^2$.
2. b) Explain why the product of the two original denominators coincides exactly with the procedure of rationalisation by the conjugate binomial.
3. c) The air safety protocol only authorises the drone to take off if the index $D$ falls within the interval $[3,40 \,;\, 3,50]$. Knowing that $1,732 < \sqrt{3} < 1,733$, bound the exact value of $D$ to three decimal places and state with justification whether flight permission is granted.