Exercises: Square root

In a square root, each number has a special name that indicates its function. Look at the following equation and identify the name of each of the elements pointed to by arrows:

$\sqrt{81} = 9$

Calculate the value of the following exact square roots mentally:

1. a) $\sqrt{16} =$
2. b) $\sqrt{4} =$
3. c) $\sqrt{100} =$
4. d) $\sqrt{49} =$
5. e) $\sqrt{1} =$
6. f) $\sqrt{121} =$

Aoife wants to decorate a square greeting card with an area of $64 \text{ cm}^2$. To buy a decorative string to go around the entire edge, she first needs to know how long one side is. What is the side length of the card?

Not all roots are exact. Without using a calculator, state between which two consecutive integers the following square roots lie:

1. a) $....... < \sqrt{20} < .......$
2. b) $....... < \sqrt{50} < .......$
3. c) $....... < \sqrt{11} < .......$

Solve the following mixed operations following the order of operations (remember to calculate the square root before addition or subtraction):

1. a) $15 + \sqrt{36} =$
2. b) $2 \times \sqrt{25} - 3 =$
3. c) $\sqrt{64} + \sqrt{9} =$
4. d) $20 - \sqrt{81} : 3 =$

A photographer wants to organise his photos on a social network to form a square grid. If he has $144$ photos in total, how many photos must he put in each row and in each column so that the grid is a perfect square?

Copy into your exercise book and use the signs $>$ (greater than), $<$ (less than) or $=$ (equal to) to compare the following numbers:

1. a) $\sqrt{36} ....... 7$
2. b) $\sqrt{81} ....... 9$
3. c) $5 ....... \sqrt{16}$
4. d) $\sqrt{100} ....... \sqrt{64}$

State whether the following statements are true or false and briefly explain why:

1. a) The square root of 25 is 5 because $5 \times 2 = 10$.
2. b) There is no integer that is the square root of 30.
3. c) The number 0 has no square root.
4. d) $\sqrt{144}$ is equal to 12 because $12^2 = 144$.

Cian has to calculate the following combined expression on an exam:

To solve it, he writes on his exam paper:

1. a) Explain in your own words what mistake Cian made and why the square root cannot be separated across an addition, unlike what happens with multiplication.
2. b) Solve the expression correctly respecting the order of operations (working out the radicand first).
3. c) Compare both results and explain whether the incorrect separation Cian did produced a result greater or smaller than the actual value.

We want to enclose a square school garden plot whose area is $75\text{ m}^2$. The side of the garden is given by $\sqrt{75}$.

1. a) Estimate: without performing divisions or using a calculator, state between which two consecutive whole numbers the length of the side must lie, identifying the two perfect squares closest to $75$.
2. b) Calculate the integer square root and the remainder of $\sqrt{75}$. If each square metre of the garden is divided into $1\text{ m}^2$ plots, what real meaning do the integer square root and the remainder have?
3. c) Check that your result is consistent using the relation $N = r^2 + \text{remainder}$ and verify that the remainder obtained is less than twice the root plus one.

Calculate the following exact square roots, justifying the result using a power, as in the example:

Example: $\sqrt{49} = 7$ because $7^2 = 49$

1. a) $\sqrt{121}$
2. b) $\sqrt{169}$
3. c) $\sqrt{225}$
4. d) $\sqrt{400}$

Find the integer square root and the remainder of the following numbers. Also state between which two consecutive natural numbers each root lies.

1. a) $\sqrt{30}$

1. b) $\sqrt{55}$

1. c) $\sqrt{85}$

Apply the properties of radicals (root of a product and root of a quotient) to solve these operations in a simplified way:

1. a) $\sqrt{64 \times 100}$
2. b) $\sqrt{16 \times 25}$
3. c) $\sqrt{\frac{144}{9}}$
4. d) $\sqrt{4 \times 9 \times 100}$

In the school grounds, a square garden has been designed with an area of $144$ $m^2$. To protect the plants, the caretaker wants to place a wooden fence all around the edge. If each metre of fencing costs $12$ €, how much will it cost to fence the entire garden?

Calculate the value of the following expressions following the order of operations:

1. a) $3^2 + \sqrt{64} - 2 \times 3$
2. b) $(\sqrt{100} - 4) : 2 + 5^2$
3. c) $2 \times \sqrt{81} - (3^2 - \sqrt{25})$
4. d) $\sqrt{49} \times 2 + 10^2 : 4$

The school photography club wants to display $200$ photos on a board, forming the largest possible square.

1. a) How many photos will each side of the square have?
2. b) How many photos will be left over?
3. c) How many more photos would be needed to complete a square with one more photo on each side?

State whether the following statements are true or false, reasoning your answer with a counterexample where necessary:

1. a) The square root of the sum of two numbers is equal to the sum of the square roots of those numbers: $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$.
2. b) The square root of a negative number does not exist.
3. c) If a number ends in $2$, it cannot be a perfect square.
4. d) The square root of a number is always less than the number itself.

Aoife has a square board with $100$ squares. Her younger brother has given her another board whose area is a quarter of the first one.

1. a) How many squares does the small board have?
2. b) How many squares long is each side of the large board?
3. c) How many squares long is each side of the small board? What is the relationship between the side length of the large board and the small one?

An orchard cooperative near Thomastown (Kilkenny) has $260$ young saplings to plant a square plot with the same number of apple trees per row as per column.

1. a) Model the situation using a square root: calculate how many apple trees each row will have in the largest square that can be formed and how many saplings will be left over.
2. b) If the cooperative wants to make the most of the order and form a square with one more row and one more column without any sapling left over, how many additional trees would they need to buy?
3. c) If the regulation spacing between every two consecutive apple trees in the same row is $6\text{ metres}$, what will be the distance from the first apple tree to the last one in a row in the planted square?

When calculating the integer square root of an unknown natural number $N$, the root obtained is $r = 11$.

1. a) Write the smallest possible value that the radicand $N$ can have so that its integer square root is still $11$.
2. b) Imagine you form a square with $11 \times 11$ tiles. Reason geometrically how many tiles you need to add at least to construct the next $12 \times 12$ square, and check that this amount equals $2 \times 11 + 1$.
3. c) From the previous deduction, determine the largest possible remainder that an integer square root with a value of $11$ can have, and deduce the largest possible value that the radicand $N$ can take.

Use the laws of indices to express the result as a single power with base 2:

Calculate the value of the following mixed expression, strictly following the order of operations:

We are designing a level for a video game on a grid. We want to build a square room using special tiles. If we use all the tiles we have, we have $15$ left over, but if we try to add one more row and one more column to make the square larger, we would be $16$ tiles short.

How many tiles do we have in total?

Find the natural number $n$ that satisfies the following equation:

Reason your answer using the concept of adding equal terms and the laws of indices.

A card collector wants to organise his album. He knows that if he arranges his cards in a square, he has $8$ cards left over. However, if he tries to form a square with one more space on each side (one more unit per side), he would be $13$ cards short.

Calculate the number of cards the collector has and the value of the integer square root of that number.

A computer virus spreads as follows: every minute, each computer that was infected in the previous minute infects $3$ other computers. If at 10:00 a.m. there is only one infected computer (the first minutes of the spread are as follows: at minute 0 there is $1$ new computer and $1$ in total; at minute 1, $3$ new and $4$ in total; at minute 2, $9$ new and $13$ in total):

1. a) Express as a power of base 3 how many computers are newly infected at 10:05.

1. b) How many computers will be infected in total at that moment (10:05)? (Remember to add the initial infected computer and those from the previous minutes.)

Find the natural number $A$ knowing that its integer square root is $25$ and that, if we add $20$ units to that number, it becomes an exact perfect square.

A courier company uses cubic boxes for its shipments. They have a large container that is also cubic and holds exactly $216$ small boxes.

1. a) How many small boxes are in the base of the container?
2. b) If they decide to manufacture a new container whose side is three times as long as the previous one, how many small boxes will fit in total in the new container? Express the result as a power.

To calculate square roots where the radicand is a product, two methods can be used:

• Method 1 (Multiply first): work out the multiplication in the radicand and then extract the square root of the resulting number.

• Method 2 (Separate factors): apply the property $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, calculating each root separately first and multiplying the results at the end.

Analyse the following two operations:

Operation A: $\sqrt{50 \times 2}$

Operation B: $\sqrt{144 \times 81}$

1. a) Try to solve Operation A using both methods. Explain why in 1st Year Method 2 is problematic for this specific case and why Method 1 is the best choice.
2. b) Solve Operation B using both methods, showing the intermediate steps. Compare the difficulty of the calculations you had to do by hand and justify which of the two methods is more advantageous in this second case.
3. c) Write a clear conclusion: under what conditions is it convenient to multiply first, and under what conditions is it convenient to separate the roots?

An artisan from Kilkenny has a box with identical square tiles to make a square decorative mural.

• If he builds the largest possible square mural with the available tiles, he has $18$ tiles left over.

• If he wanted to expand that mural to have one more row and one more column of tiles, he finds that he would be missing $15$ tiles to achieve it.

1. a) Reason geometrically how many tiles in total must be added to a square of side $r$ to turn it into one of side $r + 1$.
2. b) From the number of tiles left over and the number of tiles missing, calculate exactly how many tiles are needed to go from the first mural to the second, and deduce the value of the integer square root $r$ (the side of the original mural).
3. c) Determine exactly how many tiles the artisan has in his box and check the solution by verifying the integer square root and the remainder of that amount.

Topic challenge. Design the layout of a square study area or sustainable terrace for your home.

1. Real data collection: Use a tape measure to measure the floor of a room in your house (or check the CSO for the average floor area of a home in your region, around $90\text{ m}^2$) and choose an open area $A$ between $4\text{ m}^2$ and $16\text{ m}^2$.

2. Square root: Determine the side length of the equivalent square space by calculating $\sqrt{A}$. If it is not a perfect square, find the integer square root along with its remainder, and approximate it to one decimal place.

3. Properties of powers: You are going to cover that floor with square tiles. If each decorative tile has a side length of $2^2\text{ dm}$ and the manufacturer supplies packs of $2^3$ tiles grouped into boxes of $2^2$ packs, calculate the necessary materials by applying properties of powers (product, quotient, and power of a power). Organise and verify the calculations in a spreadsheet.

4. Final product: Write a brief report including the dimensioned floor plan, the mathematical justification, and the final order of boxes.

Assessment criteria:

• Rigorous collection of measurements and correct calculation of the square root.

• Appropriate application of power properties to simplify operations.

• Clear presentation of the report and correct use of the spreadsheet.