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) $15^2 = 225$

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

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

Marcos 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?

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.

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?

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) $\sqrt{95}$

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) $\sqrt{115}$

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 $2 \times \text{raíz} + 1$).

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 $a+1$.

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 $4.5^2 = 20.25$).

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 $90^2$).

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) $\sqrt{4 \times 81}$

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) $\sqrt{100 \times 49}$

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) $(\sqrt{100} - 4) \times 3$

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

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

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?

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?

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.

The area of a square garden is $676$ $m^2$. A security fence costing $250$ pesos per linear meter is to be installed around the garden. However, an opening of $2.5$ meters must be left for a gate on one of the sides. Work out the total budget for installing the fence.

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.