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?

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) $\sqrt{95}$

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?

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 $a+1$.

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) $\sqrt{100 \times 49}$

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 $14\text{ cm}$?

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: