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) $\sqrt{2} \cdot \sqrt{8}$
2. b) $\sqrt[3]{4} \cdot \sqrt[3]{10}$

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) $\left(\sqrt[3]{\sqrt{2}}\right)^6$

Lucía has to rationalize the fraction $\frac{6}{\sqrt[3]{2}}$ for a homework assignment. In her notebook, she carries out the following calculation:

Upon seeing that the denominator still contains a root, she adds the following comment: "It cannot be rationalized because multiplying by the same root always leaves another radical".

1. a) Identify the conceptual error Lucía makes when choosing the multiplying factor. Explain why her method works with square roots but fails with cube roots.
2. b) Based on the properties of powers, explain what exponent the radicand under the cube root needs to reach so that the radical cancels out, and determine the exact expression by which she should have multiplied the numerator and the denominator.
3. c) Correct the exercise by carrying out the rationalization and simplify the resulting fraction as much as possible.

Consider the radical expression $E = \sqrt{75} - \sqrt{12}$.

1. a) Estimate: Without using a calculator or factoring out terms, bound $\sqrt{75}$ between two consecutive integers by identifying the perfect squares surrounding it. Do the same with $\sqrt{12}$. Using these two bounds, estimate with reasoning between which two integers the value of $E$ must lie.
2. b) Calculate: Factor the radicands $75$ and $12$ into prime factors, factor out all possible terms to express both terms as like radicals, and carry out the subtraction, leaving the exact result in the form $k\sqrt{3}$ with $k \in \mathbb{N}$.
3. c) Check: Using the approximation $\sqrt{3} \approx 1,73$, find the decimal value of the result obtained in part b) and judge whether your initial estimate in part a) was correct and consistent.

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:

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

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:

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

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.

In maritime navigation, the theoretical maximum visual distance $d$ (in kilometers) at which the horizon line can be spotted from a height $h$ (in meters) above sea level is modeled by the physical formula $d = \sqrt{12,8 \cdot h}$. On the Galician coast, the visual range is evaluated from two points around the Finisterre Lighthouse: a coastal path located at $h_1 = 15\text{ m}$ above the sea and the lighthouse viewpoint located at $h_2 = 60\text{ m}$.

1. a) Substitute $h_1 = 15$ into the formula, calculate the product $12,8 \cdot 15$, factor the result into prime factors, and factor out all possible terms to give the exact distance $d_1$ in the form of a simplified radical $a\sqrt{3}\text{ km}$ ($a \in \mathbb{N}$).
2. b) Follow the same procedure for the viewpoint height $h_2 = 60$ and obtain the exact distance $d_2$ in the form $b\sqrt{3}\text{ km}$ ($b \in \mathbb{N}$).
3. c) Determine the exact visual gain of going up to the viewpoint by calculating the difference $d_2 - d_1$. Operate with the like radicals, express the final result as a single simplified radical term, and evaluate this gain in kilometers using the approximation $\sqrt{3} \approx 1,73$.

You want to design an irrational fraction of the form $\frac{A}{\sqrt{m} - \sqrt{n}}$ that simultaneously satisfies the following conditions:

1. The radicands $m$ and $n$ are natural numbers less than $10$, neither of them is a perfect square, and their difference is $m - n = 2$.

2. When rationalizing the fraction by multiplying by the conjugate of the denominator and simplifying the result as much as possible, you obtain exactly the expression $3\sqrt{5} + 3\sqrt{3}$.

1. a) Deduce with reasoning what the numbers $m$ and $n$ must be.
2. b) Set up the algebraic equality arising from the rationalization process with the values found and calculate the exact numerical value that the integer numerator $A$ must have.
3. c) Check your solution by expanding step by step the rationalization of the resulting fraction. Then, explain what result would have been obtained if the initial fraction had had a positive sign in the denominator: $\frac{A}{\sqrt{m} + \sqrt{n}}$.

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 $\sqrt{5}-1$ metres, 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 $\sqrt{6}$ metres and the two equal sides measure $\sqrt{15}$ metres each. Calculate the exact value of the height $h$ perpendicular to the base and the total area of the front wall. Rationalise the results if roots appear in the denominator.

Simplify the following expression with nested radicals, expressing the result as a single root:

$\sqrt{2 \cdot \sqrt[3]{4 \cdot \sqrt{8}}}$

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:

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

To calculate the exact value of the following combined operation with irrational fractions:

two different strategies are proposed in class:

• Method 1 (Prior rationalization): Rationalize each term separately (using the conjugate binomial in the first fraction and completing the power in the second), simplify the resulting fractions, and perform the subtraction at the end.

• Method 2 (Direct common denominator): First bring both fractions to a common denominator by multiplying the denominators together, expand the resulting numerator, and rationalize the final single fraction if necessary.

1. a) Fully work out the solution following Method 1, showing all intermediate simplifications.
2. b) Fully work out the solution following Method 2, paying attention to the products and signs in the numerator.
3. c) Compare both paths: analyze the magnitude of the numbers appearing in the intermediate steps, the complexity of the special algebraic identities, and the risk of calculation error. Justify with mathematical criteria which of the two methods is clearly more advantageous and generalizable.

A surveying team calculates the vertical design elevation $h$ (in meters) of a structure using the expression:

where $a$ and $b$ are coprime natural numbers with $a > b$, and $k \in \mathbb{N}$.

1. a) Inverse problem: When rationalizing and completely simplifying the expression for $h$, the surveying report concludes that $h = 2\sqrt{7} - 2\sqrt{3}\text{ m}$. Reconstruct the values of $a$, $b$, and $k$ in a reasoned and unique way. Justify why no other natural values satisfying the initial conditions are possible.
2. b) Operations with radicals: To reach the safety elevation, a height addition equal to $\sqrt{12}\text{ m}$ is added. Perform the addition $h + \sqrt{12}$ by factoring out and grouping like radicals, and then insert the resulting factor into the radical to express the final height as a single root of the form $\sqrt{N}$ ($N \in \mathbb{N}$).
3. c) Someone else's error: A worker claims that, since $h = \sqrt{28} - \sqrt{12}$, the initial height is $h = \sqrt{28 - 12} = \sqrt{16} = 4\text{ m}$. Explain in detail the conceptual error committed, prove through the algebraic expansion of $(\sqrt{28} - \sqrt{12})^2$ why their result is false, and state the actual approximate value of $h$ to one decimal place knowing that $\sqrt{7} \approx 2,65$ and $\sqrt{3} \approx 1,73$.