Square roots by prime factorisation

The table of squares only goes so far. What is $\sqrt{256}$, or $\sqrt{1764}$? One reli­able way is to break the num­ber into prime fac­tors and pair them up. This is called the prime fac­tori­sa­tion method. It also tells you at once whether a num­ber is a per­fect square, and what to mul­ti­ply or divide by to make it one.

Prime num­bers and prime fac­tors#

A prime num­ber is a num­ber big­ger than $1$ that can be divided exactly only by $1$ and itself. The first few primes are $2$, $3$, $5$, $7$, $11$, $13$ and so on.

Every whole num­ber big­ger than $1$ can be writ­ten as a prod­uct of primes. These primes are its prime fac­tors. To find them, keep divid­ing by the small­est prime that goes in exactly, until you reach $1$.

The method#

1. Divide the num­ber by primes, again and again, until you reach $1$.
2. Group the prime fac­tors in pairs of equal primes.
3. Take one prime from each pair.
4. Mul­ti­ply them. The answer is the square root.

Why does this work? Each pair, such as $2 \times 2$, is a square. Tak­ing one from each pair is tak­ing the square root of each square.

Exam­ple 1#

Find $\sqrt{256}$ by prime fac­tori­sa­tion.

So $256$ is eight 2s mul­ti­plied together. Put them in pairs.

$$\begin{aligned} \sqrt{256} &= \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)} \\ &= 2 \times 2 \times 2 \times 2 \\ &= 16 \end{aligned}$$

Exam­ple 2#

Find $\sqrt{1024}$.

Divid­ing by $2$ again and again: $1024$, $512$, $256$, $128$, $64$, $32$, $16$, $8$, $4$, $2$, $1$. That is ten 2s, which make five pairs.

$$\begin{aligned} \sqrt{1024} &= \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)} \\ &= 2 \times 2 \times 2 \times 2 \times 2 \\ &= 32 \end{aligned}$$

Exam­ple 3#

Find $\sqrt{1764}$.

$$\begin{aligned} \sqrt{1764} &= \sqrt{(2 \times 2) \times (3 \times 3) \times (7 \times 7)} \\ &= 2 \times 3 \times 7 \\ &= 42 \end{aligned}$$

Check: $42 \times 42 = 1764$.

Exam­ple 4#

Find $\sqrt{3136}$.

Divide by $2$ six times: $3136$, $1568$, $784$, $392$, $196$, $98$, $49$. Then $49 = 7 \times 7$.

$$\begin{aligned} \sqrt{3136} &= \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (7 \times 7)} \\ &= 2 \times 2 \times 2 \times 7 \\ &= 56 \end{aligned}$$

Exam­ple 5#

Find $\sqrt{2025}$.

$2025$ ends in $5$, so start with $5$: $2025 \div 5 = 405$, and $405 \div 5 = 81$. Then $81 = 3 \times 3 \times 3 \times 3$.

$$\begin{aligned} \sqrt{2025} &= \sqrt{(3 \times 3) \times (3 \times 3) \times (5 \times 5)} \\ &= 3 \times 3 \times 5 \\ &= 45 \end{aligned}$$

This agrees with the short cut from the last les­son.

Is it a per­fect square?#

If every prime fac­tor finds a part­ner, the num­ber is a per­fect square. If even one prime is left with­out a part­ner, the num­ber is not a per­fect square.

Exam­ple 6#

Is $2352$ a per­fect square? If not, find the small­est num­ber you must mul­ti­ply it by to make a per­fect square.

$$2352 = (2 \times 2) \times (2 \times 2) \times 3 \times (7 \times 7)$$

The $3$ has no part­ner. So $2352$ is not a per­fect square.

Give the $3$ a part­ner by mul­ti­ply­ing by $3$.

$$2352 \times 3 = 7056 = (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (7 \times 7)$$

Now every prime is paired, and $\sqrt{7056} = 2 \times 2 \times 3 \times 7 = 84$.

So the small­est num­ber to mul­ti­ply by is $3$.

Exam­ple 7#

Find the small­est num­ber by which $180$ must be divided to give a per­fect square.

$$180 = (2 \times 2) \times (3 \times 3) \times 5$$

The $5$ has no part­ner. Remove it by divid­ing by $5$.

$$180 \div 5 = 36 = (2 \times 2) \times (3 \times 3)$$

So divide by $5$, and the per­fect square is $36 = 6^2$.

Remem­ber. Pair up equal prime fac­tors and take one from each pair. A prime left with­out a part­ner means the num­ber is not a per­fect square.

Com­mon mis­takes#

• Tak­ing both primes from a pair. From $2 \times 2$ take one $2$, not both.
• Stop­ping the divi­sion too early. Keep going until you reach $1$.
• Divid­ing by a num­ber that is not prime, such as $4$ or $9$. It can still work, but you may miss a pair. Use primes.
• For­get­ting to check. Mul­ti­ply your answer by itself to see if you get the num­ber back.

Prac­tice#

1. Find $\sqrt{576}$ by prime fac­tori­sa­tion.
2. Find $\sqrt{784}$.
3. Find $\sqrt{1296}$.
4. Find $\sqrt{3969}$.
5. Find $\sqrt{4356}$.
6. Find $\sqrt{5184}$.
7. Is $98$ a per­fect square? If not, find the small­est num­ber to mul­ti­ply it by to make a per­fect square, and find the square root of the result.
8. Find the small­est num­ber by which $1152$ must be divided to give a per­fect square. Find the square root of the result.
9. Find the small­est num­ber by which $2400$ must be mul­ti­plied to give a per­fect square.
10. Find $\sqrt{6084}$.

Answers#