Arithmetic » (Square) Roots


1. The square root
2. Other roots
3. Exact or rounded off?
4. Root = power

1. The square root

It is often said that the square root is the reverse operation of squaring. That seems to be the case.
square root(25) = 5, because 5 × 5 = 25
square root(81) = 9, because 9 × 9 = 81

Why is the square root than not really the reverse of square?
Because (–5)2 = –5 × –5 is also 25, however square root(25) is always 5 (and not –5).

Example 1

You can use the square root when you need to calculate the side of a square when you know the area (A = s2).
Of a certain square the area is 56.25 cm2. The side is then square root(56.25) = 7.5 cm.

Example 2

The square root can also be used in equations: (See quadratic equations)

x2 + 5 = 25 
x2 = 20 
x = –square root(20) or x = square root(20)

Negative numbers

Finding the square root of a negative number is impossible. You cannot find a number that multiplied with itself is negative. When asked for the square root of a negative number you write down: 'not possible'.

The square root of a negative number does exist when working with complex numbers.

2. Other roots

What if you do not need the 'reverse' of a square but of a higher power.
Lets say you are wondering which number to the power of 5 is 32768.
In that case you need the fifth root of 32768. That is 8.

In mathematical language
x5 = 32768 
x  = fifth root(32768) = 8

(In this case we do not have 'or x = –8', for (–8)5 = –32768)

From (–8)5 = –32768 you can deduct that the fifth root of –32768 should be –8 agian. Roots with an odd exponent of a negative number are possible.

3. Exact or rounded off?

The square root of 7 is not a whole number. There is no whole number multiplied with itself 7.
square root(7) = 2.645751... (infinitely continuing).

When working with this number within a calculation you can work with the exact square root(7) or with the rounded 2.65. When working with the last one, you will get rounding differences and therefore an approximation. Only round off at the end of your calculation and only if the exercise asks this of you.

4. Root = power

You can also write a root as a power.
The rule is n-th root of x = x1/n.

square root(35) = 35^(1/2)
seventh root(15) = 15^(1/7)

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