Square roots

You know that $5^2 = 25$. Now turn the ques­tion round: which num­ber, mul­ti­plied by itself, gives $25$? The answer, $5$, is called the square root of $25$. Find­ing a square root is the oppo­site of squar­ing. In this les­son you will meet the square root sign, learn the square roots of the per­fect squares up to $900$, and use them in some every­day prob­lems.

What a square root is#

If $a^2 = b$, then $a$ is called the square root of $b$. We write it with the sign $\sqrt{\phantom{b}}$, like this:

$$a^2 = b \quad \Rightarrow \quad \sqrt{b} = a$$

We read $\sqrt{b}$ as "the square root of b". The arrow $\Rightarrow$ means "so" or "this tells us".

$$\begin{aligned} 1^2 = 1 &\quad \Rightarrow \quad \sqrt{1} = 1 \\ 2^2 = 4 &\quad \Rightarrow \quad \sqrt{4} = 2 \\ 3^2 = 9 &\quad \Rightarrow \quad \sqrt{9} = 3 \end{aligned}$$

A good way to think of it: the square root of a per­fect square is the num­ber that was mul­ti­plied by itself to make it.

$$\begin{aligned} \sqrt{16} &= \sqrt{4 \times 4} = 4 \\ \sqrt{25} &= \sqrt{5 \times 5} = 5 \\ \sqrt{36} &= \sqrt{6 \times 6} = 6 \end{aligned}$$

Pic­ture a square made of $36$ small tiles. Its side is $6$ tiles long. The square root of the area is the length of the side.

Square roots up to 900#

This table is the squares table from the last les­son, read back­wards.

So $\sqrt{49} = 7$, $\sqrt{144} = 12$ and $\sqrt{729} = 27$.

Remem­ber. Squar­ing and tak­ing the square root undo each other. $\sqrt{a^2} = a$ for any num­ber $a$ that is $0$ or more.

Using the pat­terns back­wards#

The pat­terns from the last les­son also work in reverse.

Mul­ti­ples of 100. Since $40^2 = 1600$, we get $\sqrt{1600} = 40$. Remove the two zeros, take the square root, and put one zero back: $\sqrt{3600} = 60$, $\sqrt{8100} = 90$.

End­ing in 25. Since $65^2 = 4225$, we get $\sqrt{4225} = 65$. To spot it, cover the $25$ and look at what is left: $42 = 6 \times 7$. So the root is $65$.

Exam­ple 1#

Find $\sqrt{2025}$.

Cover the $25$. What is left is $20$, and $20 = 4 \times 5$. So the root is $45$.

Check: $45^2 = 2025$. Yes.

Exam­ple 2#

Find $\sqrt{4900}$.

Remove two zeros to get $49$. $\sqrt{49} = 7$. Put one zero back: $70$.

$$\sqrt{4900} = 70$$

Square roots in every­day prob­lems#

Many prob­lems about squares are really ask­ing for a square root.

Exam­ple 3#

A gar­dener plants $4225$ trees. There are as many rows as there are trees in each row. How many trees are in each row?

If there are $n$ trees in each row, there are $n$ rows. So $n \times n = 4225$, and $n = \sqrt{4225}$.

$$\sqrt{4225} = 65$$

There are $65$ trees in each row.

Exam­ple 4#

Some friends go to a restau­rant. Each one spends as many rupees as there are friends. The total bill is ₹81. How many friends are there?

If there are $n$ friends, each one spends $n$ rupees. So the bill is $n \times n = 81$, and $n = \sqrt{81} = 9$.

There are $9$ friends, and each spends ₹9.

Exam­ple 5#

A square gar­den has an area of $625$ square metres. How long is each side?

The area of a square is side times side. So the side is $\sqrt{625} = 25$ m.

Num­bers that are not per­fect squares#

What about $\sqrt{50}$? Look at the squares near $50$.

$$7^2 = 49 \qquad 8^2 = 64$$

$50$ is between $49$ and $64$. So $\sqrt{50}$ is between $7$ and $8$. It is not a whole num­ber. In fact it is a lit­tle more than $7$, because $50$ is very close to $49$.

You will learn how to find such square roots as dec­i­mals in the last les­son of this chap­ter.

Com­mon mis­takes#

• Think­ing $\sqrt{16} = 8$. The square root is the num­ber times itself, not half: $4 \times 4 = 16$, so $\sqrt{16} = 4$.
• Think­ing $\sqrt{3600} = 600$. Remove two zeros and put back only one: $\sqrt{3600} = 60$.
• Mix­ing up the square and the square root. $9^2 = 81$, but $\sqrt{9} = 3$.

Prac­tice#

1. Find $\sqrt{196}$.
2. Find $\sqrt{361}$.
3. Find $\sqrt{729}$.
4. Find $\sqrt{6400}$.
5. Find $\sqrt{3025}$.
6. Find $\sqrt{7225}$.
7. A square floor tile has an area of $144$ square cen­time­tres. How long is each side?
8. An orchard has $5625$ trees, planted with as many rows as there are trees in each row. How many rows are there?
9. Each stu­dent in a class gives as many rupees as there are stu­dents. They col­lect ₹1225. How many stu­dents are in the class?
10. Between which two whole num­bers does $\sqrt{90}$ lie?

Answers#