Square roots and how to find them

If squaring takes a number and asks "what does its square look like?", the square root does the reverse: given a square, what was the original side? The symbol is $\sqrt{\,}$. So $\sqrt{49} = 7$ because $7^2 = 49$.

Concept

Formally, the (positive) square root of a non-negative number $n$ is the non-negative number $k$ for which $k \times k = n$. We write $\sqrt{n} = k$. Note that $(-7) \times (-7) = 49$ as well, but in school work the symbol $\sqrt{\,}$ always means the positive choice.

The first reliable way to find a square root is prime factorisation. Break the number into its prime factors and pair them up.

$324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 = (2 \times 3 \times 3)^2.$

Every prime appears an even number of times, so $\sqrt{324} = 2 \times 3 \times 3 = 18$. If any prime appears an odd number of times, the number is not a perfect square. For example $200 = 2^3 \times 5^2$: the $2$ appears three times, so $200$ is not a perfect square.

Prime factorisation also lets us answer the question: what is the smallest number you can multiply $n$ by to make it a perfect square? Look at the prime that has an odd power and multiply by one more copy. For $200 = 2^3 \times 5^2$, multiply by $2$: then $400 = 2^4 \times 5^2$ is a perfect square and $\sqrt{400} = 20$.

The second way is estimation between consecutive squares. Suppose you want $\sqrt{180}$. Since $13^2 = 169$ and $14^2 = 196$, we know $\sqrt{180}$ lies between $13$ and $14$. Comparing $180 - 169 = 11$ and $196 - 180 = 16$, the answer is closer to $13$, around $13.4$.

A third way is the long-division method for square roots. It looks complicated at first but it is just an organised version of "guess a digit, subtract a square, bring down the next pair". You pair the digits of the number from the right (so $5184$ becomes $51\,|\,84$), then for each pair you guess one new digit at a time.

Square roots are useful whenever a problem hides a square. Want the side of a square plot of area $625\text{ m}^2$? It is $\sqrt{625} = 25\text{ m}$. Want the time a falling ball takes? It involves $\sqrt{2h/g}$. Roots are everywhere physics meets geometry.

Worked examples

Example 1. Find $\sqrt{1296}$ using prime factorisation.

• $1296 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 = 2^4 \times 3^4$.
• Take one factor from each pair: $2 \times 2 \times 3 \times 3 = 36$.
• So $\sqrt{1296} = 36$. Check: $36^2 = 1296$. ✓

Example 2. Is $2352$ a perfect square? If not, find the smallest whole number it should be multiplied by to become one.

• $2352 = 2^4 \times 3 \times 7^2$.
• The factor $3$ appears only once , its power is odd.
• Multiply by $3$: then $2352 \times 3 = 7056 = 2^4 \times 3^2 \times 7^2$, a perfect square.
• $\sqrt{7056} = 2^2 \times 3 \times 7 = 84$.

Example 3. Estimate $\sqrt{50}$ to one decimal place.

• $7^2 = 49$ and $8^2 = 64$.
• $50$ is just one more than $49$, so $\sqrt{50}$ is just a little more than $7$.
• Try $7.1^2 = 50.41$ , too big. Try $7.07^2 \approx 49.98$ , very close.
• So $\sqrt{50} \approx 7.1$.

Example 4. A square auditorium has $441$ chairs arranged in equal rows. How many chairs per row?

• We need $\sqrt{441}$.
• $441 = 9 \times 49 = 3^2 \times 7^2$. So $\sqrt{441} = 3 \times 7 = 21$.
• $21$ chairs per row, in $21$ rows.

Try it yourself

1. Find $\sqrt{225}$.
2. Find $\sqrt{1764}$ using prime factorisation.
3. Is $980$ a perfect square? If not, by what smallest number should it be multiplied to make one?
4. Estimate $\sqrt{75}$ between two consecutive whole numbers.
5. What is the smallest perfect square divisible by $4, 9$ and $10$?
6. A square field has area $4225 \text{ m}^2$. Find its perimeter.
7. Find $\sqrt{0.81}$ and $\sqrt{0.0064}$.
8. Find a value of $n$ for which $\sqrt{n}$ is between $20$ and $21$.

Activity

Roots from prime trees. On a large sheet of paper, draw the prime-factor tree for $5184$. Circle prime factors in pairs of the same colour. Whatever colours are left over (unpaired) tell you whether the original number is a square. If everything is paired, multiply one number from each pair to read the square root straight off your drawing.