Square roots and powers are equal in the order of calculation. If there are no brackets and there are no calculations under the root entry, the powers and square roots are calculated first. If there is an addition or subtraction calculation inside the radical sign, the square roots of its components cannot be calculated separately. Therefore, the radicand must be determined by completing the calculation before the square root itself can be taken. 

Example 1

a) [[$ \sqrt {25} – \sqrt {9} = 5 – 3 = 2 $]]​

b) [[$ \sqrt {25 – 9} = \sqrt {16} = 4 $]]​

N.B! The radical sign functions like a pair of brackets. The calculations inside the radical sign must be calculated before the square root can be taken.

Example 2

Let's calculate the value of the expression [[$ \sqrt{5\cdot 3 + 3 ^ 2}$]] with a calculator.

Enter the following commands in the calculator:

This gives a square root value of [[$ 4.898979486 … ≈ 4.90 $]]​. 

In multiplications, the square root can be calculated separately from the factors of multiplication. Similarly, in division, the square root can be calculated separately from the numerator and denominator. These rules can also be applied the other way around by placing calculations under the same root entry.

Example 3

a) ​ [[$ \sqrt {4 \cdot 2} = \sqrt {4} \cdot \sqrt {2} = 2\sqrt {2} $]]​

b)  [[$ \sqrt {\displaystyle\frac {4} {9}} = \displaystyle\frac {\sqrt {4}} {\sqrt {9}} = \displaystyle\frac {2} {3} $]]​

c) [[$ \sqrt {5 \displaystyle\frac {1} {16}} = \sqrt {\displaystyle\frac {5 \cdot 16 + 1} {16}} = \sqrt {\displaystyle\frac {81} {16}} = \displaystyle\frac {\sqrt {81}} {\sqrt {16}} = \displaystyle\frac {9} {4} = 2 \displaystyle\frac {1} {4} $]]​

N.B! Mixed numbers must be converted into fractions before they can be calculated.

d) [[$ \sqrt {6^2 + 8^2} = \sqrt {36 + 64} = \sqrt {100} =10 $]]​

N.B! Because the two power calculations are connected by a plus sign, their square roots cannot be calculated separately. 

Example 4

Simplify the calculations by bringing them under the same root entry.

a) [[$ \sqrt {2} \cdot \sqrt {8} = \sqrt {2 \cdot 8} = \sqrt {16} = 4 $]]​

b) [[$ \displaystyle\frac {\sqrt {72}} {\sqrt {2}} = \sqrt {d\displaystyle\frac {72} {2}} = \sqrt {36} = 6 $]]​