Exercises: 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) $\sqrt[3]{128a^5}$

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

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

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

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) $\frac{\sqrt[3]{81}}{\sqrt[3]{3}}$

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).

Insert the factors inside the radical in the following expressions:

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

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

In an assessment test, the teacher asks to simplify the following sum of radicals:

$$\sqrt{18} + \sqrt{50} - \sqrt{8}$$

Two students submit incorrect answers:

• Seán writes: “Since the radicands are different, I apply the property of the square root of a sum and get $\sqrt{18 + 50 - 8} = \sqrt{60} = 2\sqrt{15}$”.

• Aoife states: “They are not like radicals because they have different radicands ($18 \neq 50 \neq 8$), so the expression cannot be grouped or simplified any further; it must be left with the three terms”.

1. a) Explain precisely the algebraic error made by Seán regarding the root of a sum.
2. b) Explain Aoife's conceptual error regarding the criterion needed to determine whether two or more radicals are like radicals.
3. c) Solve the expression correctly: factor out all possible factors from each root, justify how the distributive property is applied to group them, and calculate the final simplified result.

Consider the following numerical expression with radicals:

$$E = \frac{10}{\sqrt{5}} - \sqrt{45} + \frac{4}{\sqrt{5}-1}$$

1. a) Estimate: Without performing exact algebraic transformations and knowing that $2.2 < \sqrt{5} < 2.3$ (you can take $\sqrt{5} \approx 2.2$):

• Estimate the value of the first term by dividing $10$ by $2.2$.

• Estimate the value of the second term $\sqrt{45}$ knowing that it lies between $\sqrt{36} = 6$ and $\sqrt{49} = 7$.

• Estimate the value of the third term by dividing $4$ by $(\sqrt{5}-1) \approx 1.2$.

• From the three estimates, predict between which two integers the overall result of $E$ will lie.

1. b) Check and calculate: Rationalize the first term (monomial denominator), factor out terms from the second term, and rationalize the third term by multiplying by its conjugate. Group like radicals and find the simplified exact value.
2. c) Compare the exact value obtained with your previous estimate and assess the accuracy of your initial reasoning.

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

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

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:

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

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

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:

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

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

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:

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

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

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

We want to simplify the following combined operation with radicals of different indices to express it as a single irreducible radical:

$$P = \frac{\sqrt[3]{4} \times \sqrt{8}}{\sqrt[6]{32}}$$

Two classmates discuss the most suitable strategy:

• Method A (Reduction to a common index): Factor the radicands into prime factors, calculate the least common multiple of the indices, and unify everything under a single root before simplifying inside the radicand.

• Method B (Fractional exponents): Express each root as a power with base 2 and a fractional exponent, and apply the properties of the product and quotient of powers with the same base by adding and subtracting exponents.

1. a) Solve the expression step by step using Method A.
2. b) Solve the expression step by step using Method B.
3. c) Justify with reasoned arguments which of the two methods you consider more efficient and with less risk of calculation error in this specific case.
4. d) Would Method B still be as directly applicable if, instead of the number 8, you had the number 12? Justify what change would occur in the procedure with both methods.

Consider the following algebraic expression involving two unknown rational constants, $a$ and $b$:

$$K(a, b) = \frac{a}{\sqrt{3}} + \frac{b}{\sqrt{3} - 1} - \sqrt{75}$$

1. a) Rationalize the first term by removing the root from the monomial denominator.
2. b) Rationalize the second term by multiplying numerator and denominator by the conjugate of the denominator.
3. c) Factor out terms from $\sqrt{75}$ to express it as a radical like $\sqrt{3}$.
4. d) Inverse problem: Set up a system of equations and calculate the exact values of $a$ and $b$ so that two conditions are simultaneously satisfied:

1. That the irrational term in $\sqrt{3}$ cancels out completely (that is, that $K(a, b)$ is a rational number).

2. That the final numerical result of the expression is exactly equal to $4$.

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

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

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

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

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

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

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 $m^2$.

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

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

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:

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

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

• $a = \sqrt{2} + \sqrt{3}$

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

• $c = \sqrt{6}$

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

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:

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

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.

An agricultural cooperative in Navarra plans to install solar panels on two adjacent plots that share a straight boundary line:

• Plot A is square and its registered area is $72\text{ dam}^2$.

• Plot B is rectangular, has an area of $50\text{ dam}^2$, and one of its sides measures exactly $x = \frac{10}{\sqrt{2} + 1}\text{ dam}$.

1. a) Calculate the exact length of the side of plot A, simplifying the radical as much as possible by factoring out terms.
2. b) Find the exact length of the other side of plot B (denoted by $y$) by dividing its area by $x$. Rationalize the result by multiplying by the conjugate and express it in simplified form.
3. c) To optimize the fencing, the interior dividing fence (whose length coincides exactly with the shorter side of plot B) is removed, and only the joint exterior perimeter of both plots is fenced. Obtain the exact and simplified expression, without roots in denominators and grouping like terms, of the total decameters of wire fence needed.
4. d) If the cost of the fence is $18\text{ €/dam}$ and the approximation $\sqrt{2} \approx 1.414$ is used, calculate the total estimated budget and explain why it is preferable to work with the exact symbolic expression with radicals until the end rather than rounding the measurements of each side with decimals at the beginning of the project.

Consider the expression depending on two positive integer parameters $m$ and $n$:

$$T(m, n) = \frac{\sqrt[3]{2^m}}{\sqrt[4]{2^n}} \times \sqrt[12]{32}$$

1. a) Express $T(m, n)$ as a single power of base 2 with a fractional exponent in terms of $m$ and $n$, detailing the reduction of indices to a common denominator.
2. b) Knowing that $T(m, n)$ must be an integer, deduce the linear relation that $m$ and $n$ must satisfy, and find the smallest pair of strictly positive integers $(m, n)$ such that $T(m, n) = 2$.
3. c) Using the value $T(m, n) = 2$ obtained, the following combined expression with a rational parameter $k$ is defined:

$$H(k) = \frac{T(m, n)}{\sqrt{7} - \sqrt{5}} - \frac{\sqrt{175} + \sqrt{28}}{k}$$

Rationalize the first term using its conjugate, factor out terms from the second term, and determine the value of the rational number $k$ for which the coefficient of $\sqrt{7}$ cancels out, leaving $H(k)$ as a pure radical in $\sqrt{5}$.

1. d) Prove with reasoned arguments why there is no value of $k \in \mathbb{Q}$ that makes it possible to transform $H(k)$ into a non-zero rational number.

Topic challenge. How does inflation impact the domestic economy? You are going to evaluate the loss of purchasing power in Ireland and propose a household savings strategy.

1. Data collection: Access the CSO website and check the average CPI for the last year and the average annual salary (approx. $26\,900$ €). Represent the annual rate $r$ (e.g., $r = 0.032$) on the real line and calculate the absolute rounding error using $|x - \tilde{x}| \le 10^{-3}$.

2. Modelling and calculation: In a spreadsheet, model the loss in value of an initial savings amount of $10^4$ € using the function $C(t) = 10^4 \times (1 + r)^{-t}$. Express the quantities in scientific notation. Use rational exponent powers and radicals to calculate the equivalent monthly value ($t = 1/12$). Apply logarithms to accurately determine how many years it will take for the savings to be reduced by half.

3. Operations with functions and report: If household income grows according to $I(t) = I_0 \times (1 + 0.015)^t$ and the cost of living according to $G(t) = G_0 \times (1 + r)^t$, define and operate the net savings function $A(t) = I(t) - G(t)$. Write a brief report with your conclusions and recommendations.

Assessment criteria:

• Rigour in obtaining data from the CSO and bounding errors with absolute value.

• Mastery of logarithms, scientific notation, and radicals in the spreadsheet.

• Correctness when composing or operating functions, and clarity in the report's conclusions.