Square Roots
Sophia is making a square greeting card with an area of $64$ square inches. She wants to glue a ribbon along the edge, so she needs to know how long one side is. Squaring a number turns a side into an area. Which operation takes you backward, from the area to the side? It is the square root.
What Is a Square Root?
You already know that the area of a square with side $8$ in. is $8 \times 8 = 8^2 = 64$ square inches. The square root goes the other way: it starts with the area and asks for the side.
Each part of $\sqrt{81} = 9$ has a name:
Move the slider and watch how the side of the square changes with its area.
Numbers like $1, 4, 9, 16, 25, \ldots$ have a whole-number square root. They are called perfect squares, because that many tiles can be arranged into a square with none left over.
| $n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $n^2$ | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
Two special cases: $\sqrt{0} = 0$ and $\sqrt{1} = 1$. So zero does have a square root. A negative number has no square root, as the first warning below explains.
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…
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: a) $\sqrt{16} =$ b) $\sqrt{4} =$ c) $\sqrt{100} =$ d) $\sqrt{49} =$ e) $\sqrt{1} =$ f) $\sqrt{121} =$
Work out mentally the value of the following exact square roots:
- a) $\sqrt{16} =$
- b) $\sqrt{4} =$
- c) $\sqrt{100} =$
- d) $\sqrt{49} =$
- e) $\sqrt{1} =$
- f) $\sqrt{121} =$
Calculate the following exact square roots, justifying the result using a power, as in the example: Example: $\sqrt{49} = 7$ because $7^2 = 49$* a) $\sqrt{121}$ b)…
Calculate the following exact square roots, justifying the result using a power, as in the example:
Example: $\sqrt{49} = 7$ because $7^2 = 49$
- a) $\sqrt{121}$
- b) $\sqrt{169}$
- c) $\sqrt{225}$
- d) $\sqrt{400}$
Squares: Area and Side
Square roots show up whenever you know the area of a square and want its side. Remember:
What happens to the area when you change the side? Look at a square with side $5$ and a square with side $10$.
The big square holds exactly $4$ copies of the small one. If the side is multiplied by $2$, the area is multiplied by $2 \times 2 = 4$. The other way around, a square with $\tfrac{1}{4}$ of the area has half the side.
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…
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?
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…
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?
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. a) How many squares does the small…
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.
- a) How many squares does the small board have?
- b) How many squares long is the side of the large board?
- 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?
Back to Sophia's card: the side is $8$ in., so the ribbon around the edge is $4 \times 8 = 32$ in. long.
When the Root Is Not Exact
Most numbers are not perfect squares. There is no whole number that times itself gives $30$. You can still trap $30$ between two perfect squares.
To compare a root with a number, rewrite the number as a root or the root as a number. For example, $7 = \sqrt{49}$, so $\sqrt{50} > 7$. And a bigger radicand always gives a bigger root: $\sqrt{100} > \sqrt{64}$ because $100 > 64$.
Not all roots are exact. Without using a calculator, state between which two consecutive integers the following square roots lie: a) $....... < \sqrt{20} < .......$ b)…
Not all roots are exact. Without using a calculator, state between which two consecutive integers the following square roots lie:
- a) $....... < \sqrt{20} < .......$
- b) $....... < \sqrt{50} < .......$
- c) $....... < \sqrt{11} < .......$
Find the integer square root and the remainder of the following numbers. Also indicate between which two consecutive natural numbers each root lies. a) $\sqrt{30}$ b)…
Find the integer square root and the remainder of the following numbers. Also indicate between which two consecutive natural numbers each root lies.
- a) $\sqrt{30}$
- b) $\sqrt{55}$
- c) $\sqrt{85}$
A DJ wants to organize their collection of $200$ CDs on a shelf forming the largest possible square. a) How many CDs will each side of the square have? b) How many CDs…
A DJ wants to organize their collection of $200$ CDs on a shelf forming the largest possible square.
- a) How many CDs will each side of the square have?
- b) How many CDs will be left over?
- c) How many more CDs would they need to complete a square that has one more CD on each side?
Properties of Roots
Sometimes you can make a root easier by splitting it.
Apply the properties of roots (root of a product and root of a quotient) to solve these operations in a simplified way: a) $\sqrt{64 \cdot 100}$ b) $\sqrt{16 \cdot 25}$…
Apply the properties of roots (root of a product and root of a quotient) to solve these operations in a simplified way:
- a) $\sqrt{64 \cdot 100}$
- b) $\sqrt{16 \cdot 25}$
- c) $\sqrt{\frac{144}{9}}$
- d) $\sqrt{4 \cdot 9 \cdot 100}$
State whether the following statements are true or false and briefly explain why: a) The square root of 25 is 5 because $5 \cdot 2 = 10$. b) There is no integer that is…
State whether the following statements are true or false and briefly explain why:
- a) The square root of 25 is 5 because $5 \cdot 2 = 10$.
- b) There is no integer that is the square root of 30.
- c) The number 0 does not have a square root.
- d) $\sqrt{144}$ is equal to 12 because $12^2 = 144$.
State whether the following statements are true or false, justifying your answer with a counterexample when necessary: a) The square root of the sum of two numbers is…
State whether the following statements are true or false, justifying your answer with a counterexample when necessary:
- a) The square root of the sum of two numbers is equal to the sum of the square roots of those numbers: $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$.
- b) The square root of a negative number does not exist.
- c) If a number ends in $2$, it cannot be a perfect square.
- d) The square root of a number is always less than the number itself.
Order of Operations with Roots
A root works at the same level as an exponent. Do it in the "E" step of PEMDAS, before multiplying, dividing, adding or subtracting.
Remember: multiplication and division share a level, and addition and subtraction share a level. Within each level, work from left to right. Some problems write division with a colon, as in $20 : 4$. It means the same as $20 \div 4$.
The radical sign also groups what is under it: in $\sqrt{9 + 16}$, add first to get $\sqrt{25}$, then take the root.
Solve the following mixed operations respecting the order of operations (remember to calculate the square root before additions or subtractions): a) $15 + \sqrt{36} =$…
Solve the following mixed operations respecting the order of operations (remember to calculate the square root before additions or subtractions):
- a) $15 + \sqrt{36} =$
- b) $2 \cdot \sqrt{25} - 3 =$
- c) $\sqrt{64} + \sqrt{9} =$
- d) $20 - \sqrt{81} : 3 =$
Calculate the value of the following expressions, respecting the order of operations: a) $3^2 + \sqrt{64} - 2 \cdot 3$ b) $(\sqrt{100} - 4) : 2 + 5^2$ c)…
Calculate the value of the following expressions, respecting the order of operations:
- a) $3^2 + \sqrt{64} - 2 \cdot 3$
- b) $(\sqrt{100} - 4) : 2 + 5^2$
- c) $2 \cdot \sqrt{81} - (3^2 - \sqrt{25})$
- d) $\sqrt{49} \cdot 2 + 10^2 : 4$
Calculate the value of the following combined expression, strictly respecting the order of operations: $$(2^3 + \sqrt{81}) : \sqrt{25} + (5^2 - 3^3 + 2)^4 \cdot…
Calculate the value of the following combined expression, strictly respecting the order of operations:
$$(2^3 + \sqrt{81}) : \sqrt{25} + (5^2 - 3^3 + 2)^4 \cdot \sqrt{144} : 2^2$$Leftover and Short: Square Puzzles
Some puzzles say: "If I make the biggest square I can, $15$ tiles are left over. If I try a bigger square, I am short $16$ tiles." How many tiles are there?
First, a useful fact about the integer square root. If the integer square root of $N$ is $r$, then
$$r^2 \le N < (r+1)^2$$For example, if the integer square root of a number is $8$, the number is at least $8^2 = 64$ and less than $9^2 = 81$.
Now compare a square of side $n$ with the next bigger square, of side $n+1$. You must add one new row and one new column.
So in a "left over / short" puzzle, left over + short = $2n+1$, where $n$ is the side of the square that was built.
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…
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.
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 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 $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.
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.
Powers Toolkit
Square roots live next to powers: a square is the power $a^2$. Some problems need a few facts about exponents.
An exponent counts how many times the base is a factor: $2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32$. Here $2$ is the base and $5$ is the exponent.
The rules only work when the bases are the same. If they are not, rewrite them as powers of a common base. Handy ones: $4 = 2^2$, $8 = 2^3$, $16 = 2^4$, $9 = 3^2$, $27 = 3^3$, $81 = 3^4$.
Adding equal powers is different. Repeated addition is multiplication: $3^n + 3^n + 3^n = 3 \cdot 3^n = 3^{n+1}$.
Powers also describe things that grow by repeating the same step.
Find the natural number $n$ that satisfies the following equality: $$3^n + 3^n + 3^n = 3^7$$ Reason your answer using the concept of repeated addition and the properties…
Find the natural number $n$ that satisfies the following equality:
$$3^n + 3^n + 3^n = 3^7$$Reason your answer using the concept of repeated addition and the properties of powers.
A computer virus spreads in the following way: every minute, each infected computer infects $3$ other computers. If at 10:00 in the morning there is only one infected…
A computer virus spreads in the following way: every minute, each infected computer infects $3$ other computers. If at 10:00 in the morning there is only one infected computer:
- a) Express as a power of base 3 how many computers are newly infected at 10:05.
- b) How many computers will be infected in total at that time (10:05)? (Remember to add the initial infected one and those from the previous minutes).
Use the properties of powers to express the result as a single power of base 2: $$\frac{(4^3 \cdot 8^2)^2}{(2 \cdot 16^3)}$$ Hint: Express all bases as powers of 2…
Use the properties of powers to express the result as a single power of base 2:
$$\frac{(4^3 \cdot 8^2)^2}{(2 \cdot 16^3)}$$Hint: Express all bases as powers of 2 before operating.
From Squares to Cubes
A cube with edge $s$ has volume $s \cdot s \cdot s = s^3$. If you know the volume and want the edge, you ask which number multiplied by itself three times gives it. That is the cube root, written $\sqrt[3]{\ \ }$. For example, $\sqrt[3]{27} = 3$ because $3^3 = 27$.
A courier company uses cubic boxes for its deliveries. They have a large container which is also cubic and holds exactly $216$ small boxes. a) How many small boxes are…
A courier company uses cubic boxes for its deliveries. They have a large container which is also cubic and holds exactly $216$ small boxes.
- a) How many small boxes are there at the base of the container?
- 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.