Square roots

In a square root, each number has a special name that indicates its function. Look at the following equation and identify the name of each of the elements pointed to by arrows:

$\sqrt{81} = 9$

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) $\sqrt{121} =$

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

Copy into your exercise book and use the signs $>$ (greater than), $<$ (less than) or $=$ (equal to) to compare the following numbers:

1. a) $\sqrt{36} ....... 7$
2. b) $\sqrt{81} ....... 9$
3. c) $5 ....... \sqrt{16}$
4. d) $\sqrt{100} ....... \sqrt{64}$

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?

We are designing a level for a video game on a grid. We want to build a square room using special tiles. If we use all the tiles we have, we have $15$ left over, but if we try to add one more row and one more column to make the square larger, we would be $16$ tiles short.

How many tiles do we have in total?

State whether the following statements are true or false and briefly explain why:

1. a) The square root of 25 is 5 because $5 \times 2 = 10$.
2. b) There is no integer that is the square root of 30.
3. c) The number 0 has no square root.
4. d) $\sqrt{144}$ is equal to 12 because $12^2 = 144$.

Cian has to calculate the following combined expression on an exam:

$$\sqrt{64 + 36}$$

To solve it, he writes on his exam paper:

$$\sqrt{64 + 36} = \sqrt{64} + \sqrt{36} = 8 + 6 = 14$$

1. a) Explain in your own words what mistake Cian made and why the square root cannot be separated across an addition, unlike what happens with multiplication.
2. b) Solve the expression correctly respecting the order of operations (working out the radicand first).
3. c) Compare both results and explain whether the incorrect separation Cian did produced a result greater or smaller than the actual value.

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

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

1. a) $3^2 + \sqrt{64} - 2 \times 3$
2. b) $(\sqrt{100} - 4) : 2 + 5^2$
3. c) $2 \times \sqrt{81} - (3^2 - \sqrt{25})$
4. d) $\sqrt{49} \times 2 + 10^2 : 4$

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?

Calculate the value of the following mixed expression, strictly following the order of operations:

$$(2^3 + \sqrt{81}) : \sqrt{25} + (5^2 - 3^3 + 2)^4 \times \sqrt{144} : 2^2$$

Find the natural number $n$ that satisfies the following equation:

$$3^n + 3^n + 3^n = 3^7$$

Reason your answer using the concept of adding equal terms and the laws of indices.

Use the laws of indices to express the result as a single power with base 2:

$$\frac{(4^3 \times 8^2)^2}{(2 \times 16^3)}$$