Square roots of decimals and fractions

So far every square root has been a whole num­ber. But what is $\sqrt{0.01}$, or $\sqrt{1.69}$, or $\sqrt{2}$? In this les­son you will find square roots of dec­i­mals and frac­tions by turn­ing them into frac­tions with square tops and bot­toms. Then you will use long divi­sion to find square roots that never come out exactly, such as $\sqrt{2}$, cor­rect to two dec­i­mal places.

Square roots of dec­i­mals#

Change the dec­i­mal into a frac­tion over $100$, $10000$ or $1000000$. These bot­toms are per­fect squares.

$$\sqrt{100} = 10 \qquad \sqrt{10000} = 100 \qquad \sqrt{1000000} = 1000$$

Then use the rule from the last les­son: the root of a frac­tion is the root of the top over the root of the bot­tom.

$$\displaystyle \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$

Exam­ple 1#

Find $\sqrt{0.01}$.

$$\displaystyle \begin{aligned} \sqrt{0.01} &= \sqrt{\frac{1}{100}} \\ &= \frac{\sqrt{1}}{\sqrt{100}} \\ &= \frac{1}{10} = 0.1 \end{aligned}$$

Check: $0.1 \times 0.1 = 0.01$.

Exam­ple 2#

Find $\sqrt{0.0001}$.

$$\displaystyle \sqrt{0.0001} = \sqrt{\frac{1}{10000}} = \frac{\sqrt{1}}{\sqrt{10000}} = \frac{1}{100} = 0.01$$

Here $\sqrt{10000} = \sqrt{100 \times 100} = 100$.

Exam­ple 3#

Find $\sqrt{0.000016}$.

$$\displaystyle \begin{aligned} \sqrt{0.000016} &= \sqrt{\frac{16}{1000000}} \\ &= \frac{\sqrt{16}}{\sqrt{1000000}} \\ &= \frac{4}{1000} = 0.004 \end{aligned}$$

Exam­ple 4#

Find $\sqrt{1.69}$ and $\sqrt{33.64}$.

$$\displaystyle \sqrt{1.69} = \sqrt{\frac{169}{100}} = \frac{13}{10} = 1.3$$

$$\displaystyle \sqrt{33.64} = \sqrt{\frac{3364}{100}} = \frac{58}{10} = 5.8$$

For $\sqrt{3364}$ you can use long divi­sion: pairs $33 \mid 64$, then $5 \times 5 = 25$ leaves $8$, and $108 \times 8 = 864$ leaves $0$. So $\sqrt{3364} = 58$.

A quick check on the point#

Look at the exam­ples. The num­ber of dig­its after the point in the root is half the num­ber in the orig­i­nal dec­i­mal.

So a dec­i­mal needs an even num­ber of dig­its after the point before you take its root. Add a zero at the end if needed. This is why $\sqrt{0.4}$ is not $0.2$. Write $0.4$ as $0.40$: there is no whole num­ber whose square is $40$, so $\sqrt{0.4}$ does not come out exactly. It is about $0.632$. But $\sqrt{0.04} = 0.2$ exactly.

Square roots of frac­tions#

Take the root of the top and the root of the bot­tom.

Exam­ple 5#

Find $\displaystyle \sqrt{\frac{256}{441}}$.

$$\displaystyle \sqrt{\frac{256}{441}} = \frac{\sqrt{256}}{\sqrt{441}} = \frac{16}{21}$$

Exam­ple 6#

Find $\displaystyle \sqrt{16\frac{97}{144}}$.

First change the mixed num­ber into a frac­tion: $16 \times 144 + 97 = 2304 + 97 = 2401$.

$$\displaystyle \sqrt{16\frac{97}{144}} = \sqrt{\frac{2401}{144}} = \frac{49}{12} = 4\frac{1}{12}$$

Long divi­sion with dec­i­mals#

Long divi­sion also works for dec­i­mals. Pair the dig­its from the point: to the left of the point, pair going left; to the right of the point, pair going right. The point in the answer goes where the point in the num­ber is reached.

Exam­ple 7#

Find $\sqrt{477.4225}$.

Pairs: $4 \mid 77 \,.\, 42 \mid 25$.

After step 2 we cross the point, so the answer gets its point there.

$$\sqrt{477.4225} = 21.85$$

Square roots that never end#

Num­bers like $2$, $3$, $5$ and $7$ are not per­fect squares. Their square roots are irra­tional num­bers: their dec­i­mals never stop and never repeat. We can still find them to as many dec­i­mal places as we like, by long divi­sion with pairs of zeros after the point.

Exam­ple 8#

Find $\sqrt{2}$ cor­rect to two dec­i­mal places.

To be sure of two places, we work out three. Write $2$ as $2.000000$, which gives the pairs $2 \,.\, 00 \mid 00 \mid 00$.

So $\sqrt{2} = 1.414\ldots$ Some­thing is always left over, and it will never come out exactly.

Round­ing to two places. Look at the third digit after the point. If it is $5$ or more, add $1$ to the sec­ond digit. If it is less than $5$, leave the sec­ond digit as it is. Here the third digit is $4$, so

$$\sqrt{2} \approx 1.41$$

The sign $\approx$ means "is approx­i­mately equal to".

Exam­ple 9#

Find $\sqrt{5}$ cor­rect to two dec­i­mal places.

Pairs: $5 \,.\, 00 \mid 00 \mid 00$.

So $\sqrt{5} = 2.236\ldots$ The third digit is $6$, which is $5$ or more, so round up.

$$\sqrt{5} \approx 2.24$$

Remem­ber. Pair the dig­its from the point. The root has half as many dig­its after the point. A num­ber that is not a per­fect square has an irra­tional square root, which we round.

Com­mon mis­takes#

• Writ­ing $\sqrt{0.4} = 0.2$. Check: $0.2 \times 0.2 = 0.04$, not $0.4$.
• Pair­ing dec­i­mal dig­its from the right-hand end of the whole num­ber. Pair from the point, both ways.
• Round­ing from the wrong digit. To round to two places, look at the third digit.

Prac­tice#

1. Find $\sqrt{0.09}$.
2. Find $\sqrt{0.0049}$.
3. Find $\sqrt{2.25}$.
4. Find $\sqrt{12.25}$.
5. Find $\sqrt{0.000144}$.
6. Find $\displaystyle \sqrt{\frac{64}{81}}$.
7. Find $\displaystyle \sqrt{\frac{169}{196}}$.
8. Find $\sqrt{72.25}$.
9. Find $\sqrt{3}$ cor­rect to two dec­i­mal places.
10. Find $\sqrt{0.0361}$.

Answers#