maths > exponents

Representation of Squares and Square Roots

    what you'll learn...


In this page, square of a number and square root of a number are revised. The long division method to find the square root is explained.


We studied in exponents that 32=3×3=9.

An exponent of power 2 is called square of the number.

eg: Square of 7 is 72=49.

The word "square" means : the 2D shape of equal sides and 90 angles" between the sides. The exponent square is representative of "area of the shape square, side to the power 2".


We studied in the exponents that 1612=162=4".

An exponent of power 12 or the 2nd root is called square root of the number.

eg: Square root of 64 is 6412=642=8.

The power of the root need not be mentioned for square roots


Square of 4 is 42.

Square root of 9 is 92=912.


the symbol is the square root.


Square of a number : A number multiplied by itself is the square of the number.
eg: 62=6×6=36

Square Root of a number : Square root of a value is the number whose square is the given value.
eg: 36=6 as 62=36.


The factors of 26 are "1,2,13,26"

The factors of 28 are "1,2,4,7,14,28"


The prime factorization of 264 is 2×2×2×3×11.
A procedure is illustrated in the figure.

finding square roots

Consider finding the result of 3600.

Square root is a form of root. In roots, we learned to perform prime factorization to find the root.

Finding Square Roots : To find square root of a number, express the number in prime factors and group the factors.

eg: 100 =2×2×5×5 =2×5=10

Note: This method is suitable for finding square roots resulting in integers.


What is the 8
The answer is "22".

Using the prime factorization method

The prime factorization method is suitable for square roots resulting in integer values.

a procedure

Square root of a number can be given as

b2 is a 2 digit number with tens-units places
100a2 is a number that has 00 at tens-units places

This understanding gives a method to eliminate b and look at a to choose the highest digit of the square root.


The rearranged one with (20a+b) gives the method to multiply a by 2 (which is 2a) and append a value b, which is 20a+b. Then multiply b to 20a+b.

step 1
x2-a2 at 100s place=y
step 2
y-(2a joined with b)×b

In this process, the choice of a and b make the square root x=10a+b=a joined with b

The above process is explained for 2 digit square root and is easily extended for higher number of digits.

example using the procedure

square root by long division method

Consider 529

=22 100s place +(2×2 joined 3=43)×3

The above steps is used in reverse when the square root is not known
=2 at 10s place +3 at units place

This procedure is illustrated in the figure.

69169=263 square root long division


Procedure to finding Square Root of a number : Long division method is illustrated in the figure.
square root long division


The outline of material to learn "Exponents" is as follows. Note: click here for detailed outline of Exponents s

    →   Representation of Exponents

    →   Inverse of exponent : root

    →   Inverse of exponent : Logarithm

    →   Common and Natural Logarithms

    →   Exponents Arithmetics

    →   Logarithm Arithmetics

    →   Formulas

    →   Numerical Expressions

    →   PEMA / BOMA

    →   Squares and Square roots

    →   Cubes and Cube roots