Square roots

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) $\sqrt{121} =$

Lucía 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) $....... < \sqrt{11} < .......$

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

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.

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) $2 \cdot \sqrt{25} - 3 =$
3. c) $\sqrt{64} + \sqrt{9} =$
4. d) $20 - \sqrt{81} : 3 =$

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?

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?