Exercises: Square roots

Work out mentally the value of the following exact square roots using multiplication tables:

1. a) $\sqrt{36}$
2. b) $\sqrt{64}$
3. c) $\sqrt{81}$
4. d) $\sqrt{100}$
5. e) $\sqrt{121}$

Match each power with its corresponding root in your notebook, as in the example:

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

1. a) $4^2 = 16$
2. b) $9^2 = 81$
3. c) $12^2 = 144$
4. d) $15^2 = 225$

Aoife's room is square and has a total area of $16$ square metres. How long is the wall of her room?

State between which two consecutive natural numbers the following square roots lie. Look at the example:

Example: $\sqrt{10}$ lies between $3$ and $4$ because $3^2=9$ and $4^2=16$.

1. a) $\sqrt{20}$
2. b) $\sqrt{45}$
3. c) $\sqrt{70}$
4. d) $\sqrt{95}$

Apply the root of a product rule to calculate the following square roots more easily, as shown in the example:

Example: $\sqrt{4 \times 25} = \sqrt{4} \times \sqrt{25} = 2 \times 5 = 10$

1. a) $\sqrt{9 \times 100}$
2. b) $\sqrt{16 \times 25}$
3. c) $\sqrt{4 \times 81}$

Look at the following dot pattern. Which natural number represents the total number of dots? What is its square root?

Solve the following combined operations, respecting the order of operations:

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

Cian wants to arrange his collection of $40$ action figures in a square display case.

1. a) What is the largest perfect square he can form with his figures?
2. b) How many figures will be left over if he forms that square?
3. c) How many more figures would he need to form the next largest square?

When correcting a homework assignment on the board, Cian writes the following:

"Since the square root of a product can be split into the product of the square roots, the exact same thing happens with addition: $\sqrt{64 + 36} = \sqrt{64} + \sqrt{36} = 8 + 6 = 14$".

1. a) Explain in detail why Cian's reasoning is flawed, without just saying that "the rules don't allow it".
2. b) Calculate the true value of the expression $\sqrt{64 + 36}$ following the order of operations, and compare it with Cian's result.
3. c) Visualize or describe a geometric justification: can a square with an area of $100\text{ cm}^2$ have a side length of $14\text{ cm}$?

We want to find the square root of the number $85$ without using a calculator or applying the traditional square root algorithm.

1. a) Estimate the value of $\sqrt{85}$ by identifying between which two consecutive perfect squares the radicand $85$ lies, and deduce between which two consecutive natural whole numbers this value is located.
2. b) Based on your previous estimate, state the integer square root $r$ of $85$ and calculate its remainder $R$.
3. c) Check that your result is valid by verifying that the equality $N = r^2 + R$ holds and ensuring that the remainder is strictly less than twice the root plus one ($R < 2r + 1$).

A logistics company wants to store $196$ packages of medical supplies in a warehouse with a square floor plan. If the packages must be placed forming a perfect square on the floor, how many packages will be on each side? If they later receive another $30$ packages and want to form the largest possible square again, how many packages will be left over?

In a building video game, a player has $150$ stone blocks to create the base of a square tower. Work out:

1. The side length of the largest square base they can build.

2. The number of blocks left over after building it.

Work out the value of the following expressions, respecting the order of operations:

1. a) $3 \times \sqrt{81} - 2^2 \times (10 - \sqrt{49})$

1. b) $(\sqrt{100} - 2^3)^2 + \sqrt{144} : 3$

State, without using a calculator, between which two consecutive natural numbers the following square roots lie. Reason your answer based on the nearest perfect squares:

1. a) $\sqrt{45}$

1. b) $\sqrt{115}$

Use the properties of roots to solve in the simplest possible way:

1. a) $\sqrt{9 \times 16 \times 25}$

1. b) $\sqrt{3600 : 4}$

1. c) $\sqrt{100 \times 49}$

The school chess club has $85$ students enrolled. For the official year photo, the photographer wants to arrange them in a perfect square.

1. How many students will be left out of the photo if the photographer wants to form the largest possible square?

2. How many more students would be needed to add one more row and one more column to the square so that no one is left out?

It is known that a natural number has an integer square root equal to $15$ and its remainder is as large as possible. What number are we talking about?

(Hint: The remainder of a square root cannot be equal to or greater than $2 \times \text{raíz} + 1$).

Mei is designing a collage using square photos with a side length of $10\text{ cm}$. She wants to arrange them on a square piece of card with an area of $2500\text{ cm}^2$, leaving no gaps between them.

1. What is the side length of the card?

2. How many of Mei's photos fit on the card in total?

To calculate the exact value of the expression $\sqrt{144 \times 25}$, two 2nd Year students propose different strategies:

• Seán first carries out the multiplication of the radicand: $144 \times 25 = 3600$, and then calculates $\sqrt{3600}$.

• Aoife applies the distributive property of the square root over multiplication: $\sqrt{144 \times 25} = \sqrt{144} \times \sqrt{25}$, and then multiplies the roots obtained.

1. a) Fully develop both procedures until reaching the final result.
2. b) Compare the two approaches and justify which one is more advantageous if you have to do the calculation mentally or without a calculator.
3. c) If the proposed operation were $\sqrt{144 : 4}$, could Aoife's method be applied? What if it were $\sqrt{144 - 64}$? Justify in each case whether the corresponding property is applicable or not.

At an olive farm in Kilkenny, an agricultural cooperative receives a shipment of $750$ olive saplings to arrange on a plot forming a perfect square grid (the same number of trees per row as per column).

1. a) What is the largest number of olive trees that each side of the grid can have? How many saplings will be left unplanted after forming this square?
2. b) To make use of the leftover saplings and leave no gaps, they decide to buy from the nursery the exact minimum number of trees needed to form a square with one more olive tree in each row and in each column. How many additional olive trees must they buy at least?
3. c) Reason geometrically why the number of trees that must be added to go from a square of side $r$ to the next square of side $r + 1$ is always equal to $2r + 1$.

Knowing that a natural number $N$ has an integer square root $a = 45$, calculate the largest value that this number $N$ can take and what its remainder is in that case. Take into account the property that relates the remainder to the integer root so that the approximation is maximal. In the figure (not exactly to scale in the gaps) you have the square of side $a$ and the next square, of side $a+1$.

A logistics company wants to organise a fleet of $2\,500$ drones into a perfectly square formation for a night show. Due to a technical fault, $15\%$ of the drones cannot take off. Work out:

1. The side length of the largest square they can form with the remaining operational drones.

2. How many drones are left over after forming this square.

3. How many additional operational drones would be needed to add one more row and one more column to the square so that it remains complete.

Work out the value of the following combined expression, paying attention to the order of operations and the properties of roots:

The area of a square garden is $676$ $m^2$. A security fence costing $12.50$ € per linear metre is to be installed around the garden. However, an opening of $2.5$ metres must be left for a gate on one of the sides. Work out the total budget for installing the fence.

Find the value of the digit $k$ so that the number $7\,05k$ is a perfect square. Once found, calculate the square root of this number using the step-by-step algorithm. (Hint: The number is between $80^2$ and $90^2$).

A square platform has been built by joining square plastic tiles of $1$ $dm^2$ each. To edge the entire platform, $156$ blue tiles were used (forming an outer perimeter one tile wide).

1. What is the total area of the platform in $dm^2$?

2. What is the exact square root of this area?

An astronomer observes a nebula whose apparent area in the sky is approximately circular. If the calculated area is $5.76 \times 10^{10}$ $km^2$:

1. Express this quantity in standard decimal notation.

2. Calculate the radius of this nebula, knowing that the area of a circle is approximated using the formula $A \approx 3 \times r^2$ (using $\pi \approx 3$ for simplicity).

3. Express the result for the radius in scientific notation.

Determine, without using a calculator and reasoning your answer by calculating squares of nearby numbers, between which two consecutive decimal numbers with one decimal place (for example, $4.4$ and $4.5$) the square root of $75$ lies.

(Example: $\sqrt{20}$ is between $4.4$ and $4.5$ because $4.4^2 = 19.36$ and $4.5^2 = 20.25$).

We analyze the properties of the integer square root and the remainder in the set of natural numbers.

1. a) Prove with reasoning why the remainder $R$ of the integer square root $r$ of a natural number $N$ can never be greater than or equal to $2r + 1$. Support your argument with the calculation of the next perfect square, $(r + 1)^2$.
2. b) Find all two-digit natural numbers ($10 \le N \le 99$) whose integer square root is exactly equal to their remainder ($r = R$). Justify why there are no other solutions beyond the ones you found.
3. c) An unknown natural number $N$ has an integer square root $r$ and its remainder $R$ is the largest possible remainder for that root. Knowing that adding one unit makes it a perfect square ($N + 1 = 144$), calculate the exact value of $N$, its integer square root $r$, and its remainder $R$.

At the logistics hub in Ballycoolin, Dublin, a parcel delivery company stores identical square pallets on the floor, forming a large compact square block with $n$ pallets per side.

1. a) After receiving a shipment of $49$ new pallets, the supervisor verifies that they can all be placed by rearranging the load into a new compact square that has exactly one more row and one more column than the original (that is, of side $n + 1$). Set up the corresponding equation and calculate how many pallets were initially in the warehouse.
2. b) Taking the already expanded block (of side $n + 1$) as a starting point, they want to plan a future expansion so that the side becomes $n + 3$ pallets. Deduce with reasoning how many additional pallets they will need to order in total to achieve this.
3. c) If, instead of $49$ pallets, they had received only $35$ on top of the initial stock from section a), would it have been possible to expand the side of the square to $n + 1$? If not, calculate the integer square root and the remainder of the new total number of pallets, and explain what practical meaning both values have in the organization of the warehouse.