Important Formulas: Square Root & Cube Root | Quantitative Aptitude for Competitive Exams - SSC MTS / SSC GD PDF Download

What is Square Root?

The square root of a number x is a mathematical operation that provides the value which, when multiplied by itself, equals the number x. It is denoted as √x.

The square root of a number can be either positive or negative, but typically the positive square root is used. For instance, the positive square root of 9 is 3, and the negative square root of 9 is -3.

Square Root Symbol

The square root of a number x is a value that, when multiplied by itself, results in the number x. The number x is considered a perfect square.
For example:

• 2² = 4, so the square root of 4 is 2.
• 3² = 9, so the square root of 9 is 3.
• 4² = 16, so the square root of 16 is 4.

The symbol for square root is √. Therefore:

• The square root of 4 is represented as √4 = 2.
• The square root of 9 is represented as √9 = 3, and so on.

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 only for a perfect square number.
• A perfect square cannot be negative, meaning the square root of a negative number is not defined.

Square Root Formula

The square root formula is used to determine the square root of a number. The exponent formula for a number 'x' is expressed as:

If n = 2, we call it the square root. Therefore, the square root of any number 'x' can be written using the formula:

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Methods to Find Square Root

The square root of a number can be found using different methods, including the prime factorization method, repeated subtraction method, and long division method. Let's explore each of these methods in detail.

Method 1: Square Root by Prime Factorization Method

To find the square root using the prime factorization method, follow these steps:

• Step 1: Prime Factorize the Given Number: Break the number down into its prime factors.
• Step 2: Make Pairs of Same Prime Factors: Group the prime factors into pairs of identical numbers.
• Step 3: Take One Factor from Each Pair: For each pair, take one factor outside the radical sign (√).
• Step 4: Multiply All the Factors: Multiply all the factors obtained from the pairs to get the square root.

Example: Find the square root of 144.
Solution:
Steps:

• Prime Factorize the Given Number: 144 = 2 × 2 × 2 × 2 × 3 × 3 (or 144 = 2⁴ × 3²)
• Make Pairs of Same Prime Factors: (2 × 2) and (3 × 3)
• Take One Factor from Each Pair: 2 and 3
• Multiply All the Factors: 2 × 3 = 6

Result: The square root of 144 is 12.

Method 2: Square Root by Long Division Method

To find the square root using the long division method, follow these steps:

• Step 1: Place a Bar Over Every Pair of Digits: Start from the right-most side (units place) and group the digits into pairs.
• Step 2: Divide the Left-Most Number: 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 Next Pair: Bring down the next pair of digits and add the last digit of the quotient to the divisor. Find a suitable number that, when added to the divisor, creates a new divisor for the dividend.
• Step 4: Repeat the Process: Continue the process to find the quotient, which will be the square root.
• Step 5: Continue with Decimals: If necessary, add decimal places by bringing down pairs of zeros and repeating the steps.

Example: Find the square root of 625.
Solution: 
Steps:

• Place a Bar Over Every Pair of Digits: Group 62 and 5.
• Divide the Left-Most Number (62): The largest number whose square is less than or equal to 62 is 8 (since 8² = 64, but 7² = 49, which is less than 62). Start by dividing 62 by 7: 62 ÷ 7 = 8 (quotient)
• Bring Down the Next Pair (25): Now bring down the next pair (25) making the dividend 625.
• Repeat the Process: Add the quotient (8) to the divisor (7) to get 15. Divide 625 by 15, giving a quotient of 5.
• Final Result: The quotient is 25, which is the square root of 625.

Result: The square root of 625 is 25.

Method 3: Square Root by Repeated Subtraction Method

The repeated subtraction method works for perfect squares:

• Step 1: Start with the Number: Subtract consecutive odd numbers from the number starting from the given number.
• Step 2: Repeat Until You Reach Zero: Continue subtracting the next odd number until the result is zero.
• Step 3: The Number of Subtractions: The nth odd number at which the result becomes zero is the square root of the given number.

Example: Find the square root of 49.
Solution:
Steps:

• Start with the Number: Begin with 49.
• Subtract Consecutive Odd Numbers:
  49 − 1 = 48
  48 − 3 = 45
  45 − 5 = 40
  40 − 7 = 33
  33 − 9 = 24
  24 − 11 = 13
  13 − 13 = 0
• Result: The 7th odd number subtracted is 49, and the square root of 49 is 7.

Result: The square root of 49 is 7.

What is Cube Root?

The cube root of a number x is a mathematical operation that provides the value which, when multiplied by itself three times (or raised to the power of three), gives the original number x. The cube root of a number “x” is represented as ∛x.

For example, the cube root of 8 is 2, because 2 × 2 × 2 = 8, or ∛8 = 2. The cube root can also be negative for negative numbers.

Cube Root Symbol

The cube root of a number a is that number which, when multiplied by itself three times, gives the number ‘a’ itself.
For example:

• 2³ = 8, or the cube root of 8 is 2
• 3³ = 27, or the cube root of 27 is 3
• 4³ = 64, or the cube root of 64 is 4

The symbol for the cube root is 1⁄3 or ∛3.
Thus, the cube root of 8 is represented as ∛8 = 2 and that of 27 is represented as ∛27 = 3, and so on.

Properties of Cube Root

The properties of cube roots are as follows:

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

Cube Root Formula

The cube root formula is a mathematical expression used to calculate the cube root of a number. It helps in determining the cube root of any given number expressed in radical form using the symbol ∛.
Suppose x is any number such that x = y × y × y. Then, the formula for calculating the cube root is:
Cube root of x =

Where y is the cube root of the number x.

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Methods to Find Cube Root

The various methods to find the cube root include the prime factorization method, Halley's method, and others. Let's explore these methods in detail.

Method 1: Cube Root by Prime Factorization Method

To find the cube root using the prime factorization method, follow these steps:

• Step 1: Determine the prime factors of the given number.
• Step 2: For each prime factor, group them into sets of three.
• Step 3: Take one factor from each group and multiply all the factors together. If any factor cannot be grouped into sets of three, it cannot be simplified further.

Example: Find the cube root of 216. 
Solution: 

• Prime Factorization of 216: 216 = 2 × 2 × 2 × 3 × 3 × 3
• Group the prime factors in sets of three: (2 × 2 × 2) and (3 × 3 × 3) 
• Take one factor from each group: 2 and 3
• Multiply the factors: 2 × 3 = 6

Result: The cube root of 216 is 6.

Method 2: Cube Root by Halley’s Method

Halley’s method is a root-finding technique used for functions with a single real variable and a continuous second derivative.
The formula for Halley’s method is:
Where:

• a is the number whose cube root is being calculated.
• x is an initial guess for the cube root.

Example: Find the cube root of 27 using Halley’s method with an initial guess x = 3.
Solution: 
Initial guess: x = 3
Apply Halley’s method formula:

Result: The cube root of 27 is approximately 3.

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Square Root and Cube Root Table