Exercises: Operations with radicals

We are going to simplify expressions by grouping like radicals. Imagine that $\sqrt{3}$ is like an "x" or an "apple". Carry out the following additions and subtractions:

1. $5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2}$

2. $7\sqrt[3]{5} - 2\sqrt[3]{5}$

3. $4\sqrt{6} + \sqrt{6} - 8\sqrt{6}$

Use the property of the product of roots with the same index ($\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}$) to group these operations into a single root and, if possible, work out the exact final result:

• $\sqrt{2} \cdot \sqrt{32}$

• $\sqrt[3]{2} \cdot \sqrt[3]{4}$

• $\sqrt{5} \cdot \sqrt{20}$

• $\sqrt{3} \cdot \sqrt{27}$

When we have a root inside another root, we can simplify it by multiplying their indices ($\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \cdot m]{a}$). Simplify the following expressions to a single root:

• $\sqrt{\sqrt{81}}$

• $\sqrt[3]{\sqrt{64}}$

• $\sqrt{\sqrt[3]{x}}$

Sometimes the numbers inside the root have factors left over that can come out. Extract all possible factors from the following radicals.

• $\sqrt{2^5}$

• $\sqrt{3^3}$

• $\sqrt{5^4}$

• $\sqrt[3]{2^4}$

We want to fence a square plot of land with a total surface area of $225 \text{ m}^2$. To find out how many metres of fencing we need, we must first calculate the length of the side of the square.

1. Work out the length of the side ($l$) knowing that the area is $l^2 = 225$.

2. Work out the total perimeter of the plot.

You are going to buy a new television, and the dimensions are given in inches along the diagonal. The screen is rectangular, with a width of 40 cm and a height of 30 cm. Using Pythagoras' theorem ($h = \sqrt{a^2 + b^2}$), calculate the length of the screen's diagonal in centimetres.

Note: The result is an exact root.

In a 3rd year ESO exam, Elena has to simplify the operation $\sqrt{64 + 36}$. On her paper, she writes the following: "Since the square root distributes over each addend, we have $\sqrt{64 + 36} = \sqrt{64} + \sqrt{36} = 8 + 6 = 14$". Encouraged by the speed of her method, in the next exercise she tries to simplify $\sqrt{50} - \sqrt{18}$ by writing: "$\sqrt{50 - 18} = \sqrt{32}$".

1. a) Calculate the real numerical value of $\sqrt{64 + 36}$ by first carrying out the addition inside and explain why Elena's procedure proves that the square root is not distributive with respect to addition.
2. b) Point out the conceptual error she made in the second operation ($\sqrt{50} - \sqrt{18}$).
3. c) Correctly solve the operation $\sqrt{50} - \sqrt{18}$ by extracting factors from each radical until you obtain a single irreducible like radical.

We want to calculate the exact value of the expression $E = \sqrt{75} - \sqrt{27}$.

1. a) Estimate: Without using a calculator or simplifying the radicals yet, determine between which two consecutive integers $\sqrt{75}$ lies and between which ones $\sqrt{27}$ lies. From these integer approximations, give an estimated interval for the result of the subtraction $E$.
2. b) Calculate: Find the prime factorisation of the radicands, extract all possible factors from both roots, and combine like radicals to obtain the exact value in the form $k\sqrt{3}$.
3. c) Check: Taking the approximation $\sqrt{3} \approx 1,73$, find the decimal value of your result from part b) and reason whether your initial estimate was consistent with the result obtained.

Carry out the following multiplications and divisions of radicals. Remember that they must have the same index to operate. If possible, simplify the final result by extracting factors from the root.

1. a) $\sqrt{18} \cdot \sqrt{2}$
2. b) $\sqrt[3]{4} \cdot \sqrt[3]{16}$
3. c) $\frac{\sqrt{75}}{\sqrt{3}}$
4. d) $\sqrt{2x} \cdot \sqrt{8x^3}$

Extract all possible factors from the following radicals. Take into account the index of the root and the exponents of the radicands.

1. a) $\sqrt{2 \cdot 3^2 \cdot 5^5}$
2. b) $\sqrt[3]{81}$
3. c) $\sqrt{12x^3 y^4}$
4. d) $\sqrt[4]{32}$

Calculate the perimeter and area of a rectangular plot of land whose dimensions are given by radicals. The base measures $\sqrt{27}$ metres and the height $\sqrt{3}$ metres. Simplify the results as much as possible.

To add and subtract radicals, they must be like radicals (same index and same radicand). Simplify the following radicals by expressing the numbers as products of prime factors, then calculate:

1. a) $3\sqrt{8} - 5\sqrt{2} + \sqrt{50}$
2. b) $\sqrt{12} + \sqrt{75} - \sqrt{27}$
3. c) $2\sqrt[3]{16} - \sqrt[3]{54}$

We have a cubic water tank. We know that its total volume is $128 \text{ m}^3$. Calculate the length of the edge of the tank. Express the answer as a simplified radical (by taking factors out of the root) and do not use decimals.

(Hint: Remember that the volume of a cube is $V = \text{arista}^3$)

Fractional powers are another way of writing radicals. Use this relationship to simplify the following expressions and give the final answer in radical form:

1. a) Write $5^{3/4}$ in radical form.
2. b) Write $\sqrt[5]{2^3}$ in index form.
3. c) Calculate by first expressing as a power: $\sqrt[3]{\sqrt{64}}$
4. d) Simplify: $\frac{\sqrt[4]{x^3}}{\sqrt{x}}$

Apply the rules of radicals (root of a root and power of a root) to simplify the following expressions to a single irreducible radical:

1. a) $\sqrt{\sqrt[3]{64}}$
2. b) $(\sqrt[3]{2})^6$
3. c) $\sqrt{2 \cdot \sqrt{2}}$ (Hint: Move the first 2 inside the second square root)

We want to attach a guy wire to support a vertical post 4 metres high. The wire will be anchored to the ground at a distance of $\sqrt{20}$ metres from the base of the post. Calculate the exact length of cable needed using Pythagoras' theorem. Express the answer as a simplified radical.

To simplify the numerical expression $M = \dfrac{\sqrt{72} \cdot \sqrt{54}}{\sqrt{3}}$, two high school students propose different approaches:

• Marcos's method: Group the three terms under a single index-2 radical by applying the multiplication and division properties of roots with the same index ($\sqrt{\frac{72 \cdot 54}{3}}$), simplify the fraction by dividing before multiplying large numbers, and, at the end, extract factors.

• Sara's method: Decompose $72$ and $54$ into prime factors from the beginning, extract all possible factors out of each root, multiply the resulting coefficients and roots, and simplify the final quotient.

1. a) Solve the expression step by step using Marcos's method.
2. b) Solve the same expression step by step using Sara's method.
3. c) Compare both procedures: which of the two paths seemed more direct to you and carried a lower risk of making mental calculation errors? Justify your answer.

At the Parque Warner amusement park in Madrid, technicians are checking the speed reached by a free-fall gondola. According to physics models, the theoretical speed $v$ (in metres per second, $\text{m/s}$) reached by a body falling freely from a height $h$ (in metres), neglecting air resistance, is given by the formula:

$$v = \sqrt{2 \cdot g \cdot h}$$

where $g$ is the acceleration due to gravity, which for these tests is rounded to $g = 10\text{ m/s}^2$.

Two control tests are carried out: one from an intermediate height $h_1 = 20\text{ m}$ and another from the maximum height $h_2 = 45\text{ m}$.

1. a) Substitute the data and calculate the exact theoretical speeds, $v_1$ and $v_2$, extracting factors from the radicals until you obtain an integer in each case.
2. b) An operator claims: "Since the difference in heights is $45 - 20 = 25\text{ m}$, the difference between both speeds ($v_2 - v_1$) must coincide with the speed the gondola would reach falling from $25\text{ m}$". Calculate the speed corresponding to a fall from $25\text{ m}$ and, relying on the properties of radicals, justify with reasoning why the operator's claim is false.

An architectural firm has designed a decorative triangular piece for a façade. The piece is an isosceles triangle (two equal sides and one different side).

We know that the two equal sides measure $\sqrt{45}$ metres each, and the base (the unequal side) measures $\sqrt{72}$ metres.

Using Pythagoras' theorem and the rules of radicals (taking factors out and simplifying), calculate the exact height ($h$) of the triangle. Express the answer as an integer or a fully simplified radical.

Fully simplify the following numerical expression. To do this, prime factorise the radicands, take factors out of the root and work with fractions. The final answer must be expressed using a single irreducible radical.

$$ A = 2\sqrt{75} - \sqrt{\frac{147}{4}} + \frac{5}{2}\sqrt{12} - \sqrt{300} $$

We have three water tanks with different cubic shapes.

• Tank A has a capacity of $2$ cubic metres.

• Tank B has an edge length that is exactly double that of Tank A.

• Tank C has an edge length that is half that of Tank A.

• Calculate the exact edge length of Tank A, expressing it as a radical.

• Calculate the exact volume of Tank B and Tank C using index laws and rules of radicals. Do not use decimals.

Express the result of the following operation as a single power with a fractional exponent. To do this, convert all radicals to powers, apply the index laws (product and quotient of powers with the same base), and simplify the final exponent.

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

Check whether the irrational number $\phi = 1 + \sqrt{5}$ is a solution to the quadratic equation:

$$ x^2 - 2x - 4 = 0 $$

To do this, substitute $x$ with $(1 + \sqrt{5})$, expand the square of the binomial (algebraic identity) and collect like terms to see if the final result is zero.

The volume of a giant sphere used in a planetarium is $500 \pi$ cubic metres.

Given that the formula for the volume of a sphere is $V = \frac{4}{3}\pi r^3$:

1. Rearrange the formula to make the radius $r$ the subject in terms of the volume.

2. Substitute the given volume and calculate the exact value of the radius, fully simplifying the radical (taking factors out of the cube root).

Perform the following operations, grouping the terms under a single root. Remember that to multiply or divide radicals with different indices, you must first convert them to a common index (lowest common multiple of the indices).

1. a) $ \sqrt{2} \cdot \sqrt[3]{4} \cdot \sqrt[6]{8} $

1. b) $ \frac{\sqrt[4]{x^3}}{\sqrt[6]{x}} $

A bacterium is square in shape and has an area of $2,25 \cdot 10^{-10}$ square metres.

Calculate the side length of the bacterium. Express the final answer in standard form and in millimetres. (Remember that $\sqrt{a \cdot 10^n} = \sqrt{a} \cdot \sqrt{10^n}$).

An architect is designing a rectangular flowerbed for a pedestrian plaza in Valencia. The project requires the exact area to be $\sqrt{216}\text{ m}^2$ and the width to measure $\sqrt{6}\text{ m}$.

1. a) Set up the equation relating the area to the sides, solve for the length, and calculate its measure by simplifying the quotient of radicals as much as possible.
2. b) To delimit another garden area, a curb of exact length $L = 5\sqrt{2}\text{ metres}$ is ordered. The curb will be built by assembling two adjacent pieces whose measurements are given by non-exact square roots of different natural numbers, that is, $\sqrt{a} + \sqrt{b} = 5\sqrt{2}$, with $a$ and $b$ natural numbers such that $a < b$ and neither of them is a perfect square. Find all possible pairs of numbers $(a, b)$ that satisfy these conditions and justify in detail why no other solutions exist.

Consider the following equation involving nested radicals with different indices:

$$\sqrt{x \cdot \sqrt[3]{x}} \cdot \sqrt[3]{x} = 8$$

1. a) To simplify the left-hand side, two theoretical strategies can be followed:

• Strategy A (properties of radicals): Move the outer $x$ inside the cube root, apply the root of a root property, and finally multiply radicals with the same index.

• Strategy B (fractional exponents): Express each radical and factor as a power with a fractional exponent and apply the product and power of a power rules with the same base.

Choose one of the two strategies, develop it step by step, and show that the left-hand side simplifies exactly to $x$.

1. b) From the result obtained, state what value $x$ must take for the equality to hold.
2. c) A classmate claims that $x = -8$ could also be a valid solution in the real numbers because $(-8) \cdot \sqrt[3]{-8} = 16 > 0$ and its square root exists. Substitute $x = -8$ into the left-hand side of the original equation, calculate its numerical value respecting the order of operations and signs, and explain why it is ruled out as a solution to the equation.

Topic challenge. How would you optimize materials for a home renovation? Transform two rooms into equivalent square spaces by applying calculations with radicals.

1. Real-world data collection: Measure the length and width of two rooms in your home with a tape measure. In a spreadsheet, calculate their real areas $A_1$ and $A_2$ rounded to whole numbers (for example, $18\text{ m}^2$ and $8\text{ m}^2$).

2. Taking factors out: Imagine that you redistribute each room as a square surface. Set up their exact side lengths: $l_1 = \sqrt{A_1}$ and $l_2 = \sqrt{A_2}$. Decompose the radicands into prime factors and take out factors to obtain simplified radicals.

3. Operations with radicals: Calculate the total length of baseboard needed by adding the perimeters of both rooms ($4l_1 + 4l_2$), operating with like radicals in exact value. Also find the exact diagonal of the larger room ($d = l_1 \cdot \sqrt{2}$).

4. Reasoned proposal: Write a brief report with the table of measurements, the exact algebraic development, and the final approximation to two decimal places to estimate the budget for baseboards at $2,75\text{ €/m}$.

Assessment criteria:

• Rigor in real measurement and modeling of square areas.

• Skill in factorizing, taking out factors, and adding like radicals.

• Clarity of the report, contrasting the precision of exact calculation against commercial rounding.