Square Roots

Work out mentally the value of the following exact square roots using the 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) $15^2 = 225$

Look at the following mosaic formed by dots. What natural number represents the total number of dots? What is its square root?

Lucy's bedroom is square-shaped and has a total area of $16$ square feet. How long is the wall of her bedroom?

Emma is designing a collage with square photos of side $10\text{ cm}$. She wants to place them on a square piece of cardstock whose area is $2500\text{ cm}^2$ without leaving gaps between them.

1. What is the side length of the piece of cardstock?

2. How many of Emma's photos fit on the piece of cardstock in total?

The area of a square garden is $676$ $m^2$. A security fence costing $\$12,50$ per linear foot is to be installed around the garden. However, an opening of $2,5$ feet must be left for a gate on one of the sides. Calculate the total budget for the installation of the fence.

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

Example: $\sqrt{10}$ is 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) $\sqrt{95}$

Indicate, 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) $\sqrt{115}$

Determine, without using a calculator and reasoning your answer by calculating nearby perfect squares, between which two natural numbers with one decimal place the square root of $75$ lies.

(Example: $\sqrt{20}$ is between $4,4$ and $4,5$ because $4,4^2 = 19,36$ and $4,5^2 = 20,25$).

Marcus wants to organize his collection of $40$ action figures in a square display cabinet.

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 figures would he be missing to be able 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. Calculate:

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

2. The number of blocks that will be left over after construction.

A logistics company wants to store $196$ packages of medical supplies in a square-plan warehouse. If the packages must be placed forming a perfect square on the floor, how many packages will there 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?

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

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

Knowing that a natural number $N$ has an integer square root of $a = 45$, 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 maximum.

A middle school chess club has $85$ enrolled students. For the official school photo, the photographer wants to arrange them to form a perfect square.

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

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

A logistics company wants to organize a fleet of $2\,500$ drones in a perfectly square formation for a night show. Due to a technical failure, $15\%$ of the drones cannot take off. Calculate:

1. The side 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 they would need to add one more row and column to the square so that it remains complete.

A square platform has been built by joining square plastic pieces of $1$ $dm^2$ each. To border the entire platform, $156$ blue pieces have been used (forming the outer perimeter one piece wide).

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

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

Apply the property of the root of a product to calculate the following roots more simply, as shown in the model:

Model: $\sqrt{4 \cdot 25} = \sqrt{4} \cdot \sqrt{25} = 2 \cdot 5 = 10$

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

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

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

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

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

Solve the following combined operations respecting the order of operations:

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

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

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

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

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

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 $90^2$).

An astronomer observes a nebula whose apparent surface in the sky is approximately circular. If the calculated area is $5,76 \cdot 10^{10}$ $km^2$:

1. Express this amount in standard decimal notation.

2. Calculate the radius of this nebula knowing that the area of a circle is approximated by the formula $A \approx 3 \cdot r^2$ (using $\pi \approx 3$ to simplify).

3. Express the result of the radius in standard form.