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) $\sqrt{121} =$

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?

A photographer wants to organise his photos on a social network to form a square grid. If he has $144$ photos in total, how many photos must he put in each row and in each column so that the grid is a perfect square?

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) $\sqrt{100} ....... \sqrt{64}$

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 $12^2 = 144$.

Calculate the following exact square roots, justifying the result using a power, as in the example:

Example: $\sqrt{49} = 7$ because $7^2 = 49$

1. a) $\sqrt{121}$
2. b) $\sqrt{169}$
3. c) $\sqrt{225}$
4. d) $\sqrt{400}$

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) $\sqrt{85}$

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?

A card collector wants to organise his album. He knows that if he arranges his cards in a square, he has $8$ cards left over. However, if he tries to form a square with one more space on each side (one more unit per side), he would be $13$ cards short.

Calculate the number of cards the collector has and the value of the integer square root of that number.

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

Apply the properties of radicals (root of a product and root of a quotient) to solve these operations in a simplified way:

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

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.

Marta has a square board with $100$ squares. Her younger brother has given her another board whose area is a quarter of the first one.

1. a) How many squares does the small board have?
2. b) How many squares long is each side of the large board?
3. c) How many squares long is each side of the small board? What is the relationship between the side length of the large board and the small one?

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) $20 - \sqrt{81} : 3 =$

In the school grounds, a square garden has been designed with an area of $144$ $m^2$. To protect the plants, the custodian wants to place a wooden fence all around the edge. If each meter of fencing costs $720$ pesos, how much will it cost to fence the entire garden?

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) $\sqrt{49} \times 2 + 10^2 : 4$

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

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?

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

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

A computer virus spreads as follows: every minute, each computer that was infected in the previous minute infects $3$ other computers. If at 10:00 a.m. there is only one infected computer (the first minutes of the spread are as follows: at minute 0 there is $1$ new computer and $1$ in total; at minute 1, $3$ new and $4$ in total; at minute 2, $9$ new and $13$ in total):

1. a) Express as a power of base 3 how many computers are newly infected at 10:05.

1. b) How many computers will be infected in total at that moment (10:05)? (Remember to add the initial infected computer and those from the previous minutes.)

Find the natural number $A$ knowing that its integer square root is $25$ and that, if we add $20$ units to that number, it becomes an exact perfect square.

A courier company uses cubic boxes for its shipments. They have a large container that is also cubic and holds exactly $216$ small boxes.

1. a) How many small boxes are in the base of the container?
2. b) If they decide to manufacture a new container whose side is three times as long as the previous one, how many small boxes will fit in total in the new container? Express the result as a power.

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