Exercises: Radicals

Express as a power with a fractional index and simplify fully the expression $\sqrt[6]{x^4 y^2 z^8}$, assuming that all variables represent positive real numbers.

Take out as many factors as possible from inside the following radical, simplifying the expression fully: $\sqrt[3]{108 a^5 b^7 c^3}$.

Express the radicals $\sqrt[3]{5}$, $\sqrt[4]{10}$ and $\sqrt[6]{30}$ with a common index, and order them from smallest to largest, justifying your answer.

Perform the additions and subtractions of the following like radicals, first extracting factors to simplify the result: $5\sqrt{18} - 2\sqrt{50} + \sqrt{8}$.

Multiply the following radicals with the same index and extract any possible factors from the final result: $\sqrt[4]{27a^2b^3} \cdot \sqrt[4]{3a^3b}$.

Calculate the power of the following radical and simplify the result by extracting all possible factors (assume positive variables): $(\sqrt[4]{8x^3y^2})^2$.

Simplify the following expression by applying the root of a root property and expressing the result with a single radical: $\sqrt[3]{a^2 \sqrt{a}}$.

Determine whether the radicals $\sqrt[4]{25x^2}$ and $\sqrt{5x}$ are equivalent for any value of $x > 0$. Justify your method using fractional indices.

In a Mathematics I exam, Marcos has to rationalize the expression $\frac{6}{\sqrt[3]{2}}$. To solve it, he writes the following procedure on his exam:

Seeing that a root still remains in the denominator, Marcos concludes: "This fraction cannot be rationalized because the index is 3 and multiplying the root by itself never eliminates the radical."

1. a) Identify the conceptual mistake Marcos made when choosing the rationalizing factor and explain which power property he forgot to take into account regarding the required exponent inside the radicand.
2. b) Correct the exercise step by step, showing how it should be properly rationalized and simplifying the final result as much as possible.

Consider the real number $N = \frac{1}{\sqrt{10} - 3}$.

1. a) Without using a calculator or previously calculating exact decimals, reasonably estimate the order of magnitude of $N$: knowing that $3 < \sqrt{10} < 3,2$, reason whether the denominator is a number close to zero or greater than 1, and deduce whether the value of $N$ will be a small number or noticeably greater than 1.
2. b) Rationalize the expression of $N$ by multiplying by the conjugate of the denominator and completely simplify the result.
3. c) Taking the approximation $\sqrt{10} \approx 3,16$, find the decimal value of the simplified expression and check whether it confirms your initial estimate from part a).

Rationalise the denominator of the following expression and simplify the final result, extracting factors if possible: $\frac{6x}{\sqrt[5]{8x^2y^3}}$.

Rationalise the denominator by multiplying by the conjugate expression and simplify by expanding the numerator: $\frac{2\sqrt{3} - \sqrt{2}}{\sqrt{3} + 2\sqrt{2}}$.

Carry out the following mixed operation with radicals of different indices, expressing the final result as a single simplified radical: $\frac{\sqrt{2x} \cdot \sqrt[3]{4x^2}}{\sqrt[6]{8x^5}}$.

The period $T$ of a simple pendulum is given by the physics formula $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length of the string and $g$ is acceleration due to gravity. If an engineer needs the new period to be exactly $\sqrt{2} + 1$ times the original period, by what exact factor must they multiply the original length $L$? Express your reasoning algebraically by expanding the square of the binomial.

In a rectangular prism, the lengths of its three concurrent edges measure exactly $\sqrt{2}$ m, $\sqrt{3}$ m and $\sqrt{5}$ m. Calculate the exact value of the space diagonal of the prism ($D = \sqrt{a^2 + b^2 + c^2}$) and its volume, expressing both results in their simplest radical form.

The annual compound interest rate $r$ (expressed as a decimal) required for an initial principal $C_0$ to grow into a final amount $C_n$ after $n$ years is given by $r = \sqrt[n]{\frac{C_n}{C_0}} - 1$. Calculate the exact value of the rate $r$ if an investment multiplies its value by $\frac{5}{2}$ in exactly 3 years. Rationalise the resulting radical in your answer.

Expand the following expression using algebraic identities for the square of a difference and the product of binomials, and collect like terms: $(\sqrt{5} - 2\sqrt{3})^2 - (2\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})$.

According to Kleiber's Law in biology, the basal metabolic rate $R$ of a mammal scales with its body mass $M$ according to the relationship $R = k \sqrt[4]{M^3}$, where $k$ is a constant. If animal $A$ has a mass exactly $27$ times greater than that of animal $B$, determine the exact ratio $\frac{R_A}{R_B}$ using properties of radicals and extracting factors.

The goal is to construct a fraction with radicals of the form $\frac{k}{\sqrt{a} + \sqrt{b}}$, where $a$, $b$, and $k$ are positive integers, with $a > b$ and where neither of the two radicands is a perfect square. It is required that, after rationalizing and fully simplifying, the result obtained is exactly $\sqrt{7} - \sqrt{3}$.

1. a) Algebraically develop the rationalization of the general expression by multiplying by the conjugate, and deduce what exact numerical relationship must exist between $k$, $a$, and $b$ so that the denominator completely disappears and the numerator matches the required result.
2. b) Provide a specific triplet of integers $(a, b, k)$ that satisfies all the conditions of the statement.
3. c) Justify in a reasoned manner whether the solution $(a, b, k)$ you gave is unique or if there are infinitely many integer triplets that satisfy this condition. If there are infinitely many, describe the rule that relates all of them.

In an experimental small wind energy facility in Albacete, the electrical power $P$ (in watts, $\text{W}$) generated by a wind turbine is modeled by the formula $P = \frac{1}{2} \cdot \rho \cdot S \cdot v^3$, where $\rho$ is the air density in the area ($\rho = 1,2\text{ kg/m}^3$), $S$ is the circular area swept by the blades in $\text{m}^2$ ($S = \pi \cdot R^2$), and $v$ is the wind speed in $\text{m/s}$.

The technical data of the prototype indicate that the radius of its blades is $R = \sqrt[4]{125}\text{ m}$ and the anemometers record an average wind speed of $v = \sqrt[6]{125}\text{ m/s}$.

1. a) Express both the radius $R$ and the speed $v$ as powers of base 5 with irreducible fractional exponents.
2. b) Express the density $\rho = 1,2$ as an irreducible fraction and substitute all terms into the power formula. Apply exclusively the properties of powers with the same base to simplify the expression and obtain the exact value of $P$ as an integer multiplied by $\pi$, without radicals or denominators.
3. c) Knowing that $1\text{ kW} = 1000\text{ W}$, calculate the generated power in kilowatts taking $\pi \approx 3,14$ and interpret whether the result has a reasonable order of magnitude for a household device.

Simplify fully the following expression with nested radicals, reducing it to a single radical and extracting all possible factors (assume that the variables are positive reals):

$\sqrt[4]{\frac{x^3 y}{z} \sqrt[3]{\frac{z^2}{x y^2} \sqrt{\frac{x}{y}}}}$.

Rationalise the denominator of the following fraction containing three radical terms and simplify the final result: $\frac{4}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$.

Prove algebraically that the irrational number $x = \sqrt{7 + 4\sqrt{3}} + \sqrt{7 - 4\sqrt{3}}$ is actually an integer. Set up the equation $x^2$ and use algebraic identities to solve it.

In a regular pentagon of side length $L = 2$ cm, the length of any diagonal $D$ is related to the side length by the golden ratio $\Phi = \frac{1+\sqrt{5}}{2}$, satisfying $D = L \cdot \Phi$. Draw a horizontal diagonal to divide the pentagon into an upper isosceles triangle and a lower isosceles trapezium. Using Pythagoras' theorem, calculate the exact value of the total height $h$ of the pentagon by adding the heights of both shapes. Express the final result in the simplest possible form.

You can do it on paper and send a photo with your answer.

The Lorentz factor in special relativity theory is defined as $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$. If a subatomic particle travels at a speed of $v = \frac{\sqrt{5}}{3}c$, calculate the exact value of $\gamma$. Subsequently, determine the exact speed $v$ (in terms of $c$) at which it must travel for the factor to be $\gamma = \sqrt{2} + 1$, rationalising the result if necessary.

The maximum stress $S$ supported by the base of an architectural dome modelled using a conic section is given by the expression $S = \frac{P}{\sqrt{a^2 + b^2 - a\sqrt{b}}}$. If the structural load is $P = 14$ tonnes, with geometric parameters $a = \sqrt{2}$ and $b = 8$, calculate the exact value of the stress $S$, rationalising the final result.

Simplify the following algebraic expression completely, performing operations on the fractions using a common denominator and applying the properties of conjugate radicals:

$\frac{\sqrt{x+1} - \sqrt{x-1}}{\sqrt{x+1} + \sqrt{x-1}} + \frac{\sqrt{x+1} + \sqrt{x-1}}{\sqrt{x+1} - \sqrt{x-1}}$.

Consider the Spiral of Theodorus, constructed using adjacent right-angled triangles. The first triangle has equal legs of length $a$. The nth triangle has legs given by the hypotenuse of the previous triangle and a new leg of length $a$. If the area of the sixth triangle in the sequence is exactly $A_6 = 12\sqrt{6}$ cm², calculate the exact value of the parameter $a$ and, with it, determine the exact perimeter of the third triangle in the spiral, simplifying the radicals.

To calculate the numerical value of the expression $E = \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} - \sqrt{2}} + \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}}$, two algebraic strategies can be followed:

• Method 1 (Individual rationalization): Rationalize each term separately by multiplying each numerator and denominator by the conjugate of its own denominator, and add the resulting expressions at the end.

• Method 2 (Direct common denominator): Treat the expression as an addition of fractions with different denominators, calculating the common denominator directly by multiplying both denominators and cross-multiplying the numerators.

1. a) Fully develop both methods step by step and verify that they yield exactly the same integer.
2. b) Compare both approaches: which of the two methods is more advantageous and minimizes the number of binomial products? Justify your choice by analyzing the structure of the denominators involved.
3. c) Use the method you selected as most efficient in part b) to generalize the calculation: deduce a simplified expression for $S(a, b) = \frac{\sqrt{a} + \sqrt{b}}{\sqrt{a} - \sqrt{b}} + \frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}}$ (with $a, b > 0$ and $a \neq b$), presenting it as a single algebraic fraction without roots.

The UNE-EN ISO 216 standard, used in Spain for the standardization of paper sizes (DIN series), establishes two conditions for the base sheet $\text{DIN A0}$: its total area must be exactly $1\text{ m}^2$, and if the sheet is cut in half parallel to its shorter side, the two resulting rectangles must retain exactly the same aspect ratio (geometric similarity) as the initial sheet.

1. a) Letting $x$ be the shorter side and $y$ be the longer side of a sheet ($y > x$), algebraically prove that the similarity condition after cutting uniquely requires the ratio between the sides to be $\frac{y}{x} = \sqrt{2}$. Explain why a standard rational proportion (such as $1,5$) would prevent preserving the shape of the document when scaling it down.
2. b) From the system formed by $\frac{y}{x} = \sqrt{2}$ and the area $x \cdot y = 1$, algebraically deduce that the exact width of the $\text{DIN A0}$ format is $x = \frac{1}{\sqrt[4]{2}}\text{ m}$ and its length is $y = \sqrt[4]{2}\text{ m}$. Rationalize the expression for the width $x$ to express it with a radical in the numerator and an integer in the denominator.
3. c) Knowing that a standard $\text{DIN A4}$ sheet is obtained by folding the $\text{DIN A0}$ sheet four consecutive times, its dimensions are $x_4 = \frac{x}{4}$ and $y_4 = \frac{y}{4}$. Obtain the exact expression for the perimeter of a $\text{DIN A4}$ sheet in meters, presenting it as a single rationalized fraction with an integer denominator and whose numerator has the form $a\sqrt[4]{b} + c\sqrt[4]{d}$, having extracted all possible factors.

Topic Challenge. Energy and electrical grid audit.

1. Data and sets: Go to the Red Eléctrica de España website (REE-RedOS) and extract the current peninsular electricity demand (in MW). Express the value in W using scientific notation, place it on the real line, and classify it into number sets ($\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$).

2. Errors and absolute value: Round the average demand to two significant figures and find the absolute and relative errors made. Using an absolute value inequality, $|x - x_0| \le 0,05 \cdot x_0$, set up the $5\,\%$ tolerance interval allowable for the grid, representing it on the real line.

3. Complex numbers and rational function: In GeoGebra, calculate the total impedance of a transformer operating in polar form: multiply $Z_1 = 15_{45^\circ}\,\Omega$ by $Z_2 = 2_{-15^\circ}\,\Omega$ and divide the result by $Z_3 = 6_{30^\circ}\,\Omega$. Finally, graph the electric current as a function of a variable resistance $R$ using the rational function $I(R) = \frac{230}{R + 4,5}$ and describe its asymptotes.

Write a short report with your screenshots, calculations, and conclusions.

Assessment criteria:

• Rigour in scientific notation, classification into sets, and error calculation.

• Correct modeling of the tolerance interval using the absolute value function.

• Precision in complex algebra in polar form and graphical analysis of the rational function.