Exercises: Square root

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 indicated by arrows:

$\sqrt{81} = 9$

Work out mentally the value of the following exact square roots:

1. a) $\sqrt{16} =$
2. b) $\sqrt{4} =$
3. c) $\sqrt{100} =$
4. d) $\sqrt{49} =$
5. e) $\sqrt{1} =$
6. f) $\sqrt{121} =$

Sophia wants to decorate a square greeting card that has 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 length of the side 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) $....... < \sqrt{11} < .......$

Solve the following mixed operations respecting the order of operations (remember to calculate the square root before additions or subtractions):

1. a) $15 + \sqrt{36} =$
2. b) $2 \cdot \sqrt{25} - 3 =$
3. c) $\sqrt{64} + \sqrt{9} =$
4. d) $20 - \sqrt{81} : 3 =$

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

Copy into your notebook 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 $5 \cdot 2 = 10$.
2. b) There is no integer that is the square root of 30.
3. c) The number 0 does not have a 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 indicate between which two consecutive natural numbers each root lies.

1. a) $\sqrt{30}$
2. b) $\sqrt{55}$
3. c) $\sqrt{85}$

Apply the properties of roots (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) $\sqrt{4 \cdot 9 \cdot 100}$

In the school courtyard, 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 around the entire edge. If each foot of fence costs $\$12$, how much will it cost to fence the whole garden?

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

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

A DJ wants to organize their collection of $200$ CDs on a shelf forming the largest possible square.

1. a) How many CDs will each side of the square have?
2. b) How many CDs will be left over?
3. c) How many more CDs would they need to complete a square that has one more CD on each side?

State whether the following statements are true or false, justifying your answer with a counterexample when 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.

Olivia has a square board with $100$ squares. Her little 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 the side of the large board?
3. c) How many squares long is the side of the small board? What is the relationship between the side of the large board and the side of the small one?

Use the properties of powers to express the result as a single power of base 2:

Hint: Express all bases as powers of 2 before operating.

Calculate the value of the following combined expression, strictly respecting 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 short of $16$ tiles.

How many tiles do we have in total?

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

Reason your answer using the concept of repeated addition and the properties of powers.

A card collector wants to organize their album. They know that if they place their cards forming a square, they have $8$ cards left over. However, if they try to form a square with one more space on each side (one more unit per side), they would be short of $13$ cards.

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

A computer virus spreads in the following way: every minute, each infected computer infects $3$ other computers. If at 10:00 in the morning there is only one infected computer:

1. a) Express as a power of base 3 how many computers are newly infected at 10:05.
2. b) How many computers will be infected in total at that time (10:05)? (Remember to add the initial infected one 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 this number, it becomes an exact perfect square.

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

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