Important Formulas: Square Root and Cube Root | Quantitative Aptitude (Quant) - CAT PDF Download

How to Find Square Root and Cube Root?

To find the square root of any number, we need to find a number that when multiplied twice by itself gives the original number. Similarly, to find the cube root of any number we need to find a number that when multiplied three times by itself gives the original number. 

Methods to Find Square Root

The different methods to find the square root of a number are prime factorization method, repeated subtraction method, long division method.

Method 1: Square Root by Prime Factorization Method
To find the square root by the prime factorization method, follow the below steps:

• Step 1: Prime factorize the given number.
• Step 2: Make pairs of the same prime factors.
• Step 3: For each pair, take that factor once out of the radical sign.
• Step 4: Now, multiply all the factors to obtain the square root.

Method 2: Square Root by Long Division Method
To find the square root by the long division method, follow the below steps:

• Step 1: Place a bar over every pair of digits of the number starting from the units' place (right-most side).
• Step 2: We divide the left-most number by the largest number whose square is less than or equal to the number in the left-most pair.
• Step 3: Bring down the number under the next bar to the right of the remainder. Add the last digit of the quotient to the divisor. To the right of the obtained sum, find a suitable number that, together with the result of the sum, forms a new divisor for the new dividend that is carried down.
• Step 4: The new number in the quotient will have the same number as selected in the divisor. The condition is the same as being either less than or equal to the dividend.
• Step 5: Now, we will continue this process further using a decimal point and adding zeros in pairs to the remainder. 
• Step 6: The quotient thus obtained will be the square root of the number. 

Method 3: Square Root by Repeated Subtraction Method

Square of any perfect square number can be determined by the repeated subtraction method. In this method, we have to repeatedly subtract the given number by consecutive odd numbers until we get the answer zero. The nth odd number for which we get the answer zero is the square root of the given number.

Methods to Find Cube Root

The different methods to find the cube root are prime factorization method, and halley’s method, etc. Let us learn these methods in detail.

Method 1: Cube Root by Prime Factorization Method

• Step 1: Determine the prime factors of a given number.
• Step 2: For every prime factor in the prime factorization, create a group of three
• Step 3: Take that factor once for each group before multiplying all the prime factors. If a factor cannot be divided into three, it cannot be further simplified.

Method 2: Cube Root by Halley’s Method
A root-finding algorithm called Halley’s method is used for functions with a single real variable with a continuous second derivative.
The formula of Halley’s Method is as follows:

where, a = number whose cube root is being calculated, and
x = integer guess of its cube root.

Square Root and Cube Root Table

Remembering the values of square root and cube root will be helpful and good for faster maths calculations. Following is the square root and cube root list of the first 20 natural numbers.

Square Root and Cube Root Formula

The square root and cube root formula is explained below in detail. Square Root Formula: The square root formula is used to find the square root of a number. We know the exponent formula for a number ‘x ’ is given as
If n=2 , we call it square root. So, the square root of any number ‘x ’, can be expressed using the formula:

Cube Root Formula: The cube root formula is a mathematical formula that gives the cube root of a number. The cube root formula helps in the calculation of the cube root of any given number that is expressed in a radical form using the symbol 3√ .
Suppose, x is any number such that, x = y × y × y . Then the formula to calculate the cube root is given as:

where, y is the cube root of any number x .

Properties of Square Root

The properties of square roots are as follows:

• The square root of a perfect square is a rational number.
• The square root of a non-perfect square is an irrational number.
• The square root of an even perfect square is even.
• The square root of an odd perfect square is odd.
• A perfect square root exists for a perfect square number only.
• A perfect square cannot be negative and hence the square root of a negative number is not defined.

Properties of Cube Root

The properties of cube roots are as follows:

• The cube root of all the odd numbers is an odd number.
• Cube root of all even natural numbers is even.
• The cube root of a negative integer always results in negative.
• The cube root of a perfect cube is a rational number.
• The cube root of a non-perfect cube is an irrational number.

Square Root and Cube Root Solved Examples

Q1: Find the square root of 36 using the prime factorization method.
Sol: Resolving 36 as the product of primes, we get

Q2: Find the square root of 16 by repeated subtraction method.
Sol: For the given number 16 , steps for repeated subtraction are as follows:
Step 1: 16−1=15
Step 2: 15−3=12
Step 3: 12−5=7
Step 4: 7−7=0
Since, the result zero is obtained in the 4th step, this means that the square root of 16 is 4 .