In this article
What is the square root of 72? A calculator says 8.485281…, which is hardly useful in an exam or a drawing. In exact form it is 6√2, and once you see why, you can simplify any square root yourself.
What a root is
Roots are the reverse of powers. Squaring 7 gives 49, so the square root of 49 is 7. Cubing 3 gives 27, so the cube root of 27 is 3. In general, the nth root of x is the number that you multiply by itself n times to get x. It is written ⁿ√x, and it equals x^(1/n).
Perfect squares you should know
A perfect square is a whole number whose square root is also a whole number. Knowing the first fifteen makes estimating and simplifying much faster.
| n | n² | n | n² | n | n² |
|---|---|---|---|---|---|
| 1 | 1 | 6 | 36 | 11 | 121 |
| 2 | 4 | 7 | 49 | 12 | 144 |
| 3 | 9 | 8 | 64 | 13 | 169 |
| 4 | 16 | 9 | 81 | 14 | 196 |
| 5 | 25 | 10 | 100 | 15 | 225 |
The square root of any number in the n² column is the number beside it. √169 = 13.
Perfect cubes follow the same idea: 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000 are the cubes of 1 to 10.
How to estimate a square root by hand
Say you need √50. Find the two perfect squares around it: 49 (which is 7²) and 64 (which is 8²). So √50 sits between 7 and 8, and because 50 is only 1 above 49, it is very close to 7. A good estimate is 7.07, and the exact value is 7.0711 to four decimal places.
A refinement that works well: guess g, then average g with x ÷ g. For √50 start with 7. Then 50 ÷ 7 = 7.143 and the average of 7 and 7.143 is 7.071, already accurate to three decimals.
How to simplify a square root
A square root is in simplest form when no perfect square (other than 1) divides the number inside. The method uses prime factors.
- Break the number into prime factors. 72 = 2 × 2 × 2 × 3 × 3.
- Group the factors in identical pairs. Here: (2 × 2), (3 × 3), and one 2 left over.
- Move one number from each pair outside the root, and keep the leftovers inside. Outside: 2 × 3 = 6. Inside: 2.
- Write the result: √72 = 6√2.
| Number | Largest square factor | Simplified | Decimal |
|---|---|---|---|
| √50 | 25 | 5√2 | 7.0711 |
| √72 | 36 | 6√2 | 8.4853 |
| √98 | 49 | 7√2 | 9.8995 |
| √200 | 100 | 10√2 | 14.1421 |
| √45 | 9 | 3√5 | 6.7082 |
| √17 | 1 | √17 | 4.1231 |
√17 cannot be simplified because 17 is prime. The calculator shows this as √17.
Cube roots and higher roots
The same idea extends to any degree. The cube root of 64 is 4, because 4 × 4 × 4 = 64. The fourth root of 81 is 3, because 3 × 3 × 3 × 3 = 81. For numbers that are not perfect powers you get a decimal, such as the cube root of 2, which is about 1.2599.
Negative numbers. An odd root of a negative number is fine: the cube root of −8 is −2, because (−2) × (−2) × (−2) = −8. An even root of a negative number has no real answer, since no real number multiplied by itself an even number of times is negative. That is why √−4 shows an error in a real-number calculator.
Rules that save time
- Product rule: √(a × b) = √a × √b. This is what makes √72 = √36 × √2 = 6√2 work.
- Quotient rule: √(a ÷ b) = √a ÷ √b. So √(9 ÷ 4) = 3 ÷ 2 = 1.5.
- Power link: the nth root of x equals x^(1/n), so √x = x^0.5.
- A warning: √(a + b) is not √a + √b. √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5.
Where roots show up
The diagonal of a square with side 10 cm is 10√2, about 14.14 cm. The side of a square with area 150 m² is √150, about 12.25 m. The Pythagorean theorem for a 5 and 12 right triangle gives a hypotenuse of √(25 + 144) = √169 = 13. Standard deviation in statistics is a square root, and so are many physics formulas for speed and frequency.
Questions people ask
How do I find the square root of a number?
Find the number that multiplied by itself gives your number. √49 = 7 because 7 × 7 = 49. For non-perfect squares, estimate between two perfect squares or use a calculator.
How do I simplify √72?
Factor 72 as 36 × 2, take the square root of 36 (which is 6) outside the radical, and you get 6√2.
What is the cube root of 27?
3, because 3 × 3 × 3 = 27.
Can you take the square root of a negative number?
Not in the real numbers. No real number squared is negative. The result is an imaginary number such as √−4 = 2i, which a real-number calculator does not show.
What is the difference between a root and a power?
They are inverses. Squaring 7 gives 49, and the square root of 49 gives 7 back. The nth root of x is the same as x^(1/n).
Is √(a + b) equal to √a + √b?
No. √9 + √16 = 7 but √(9 + 16) = √25 = 5. Roots do not split over addition, only over multiplication and division.