Square Root

Understanding square roots is fundamental in mathematics, especially for competitive exams like RRB Group D. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. We denote the square root symbol as '√'. So, √9 = 3.

Definition and Notation

Mathematically, if 'a' is a number, its square root 'b' is such that b² = a. Every positive number has two square roots: a positive one (called the principal square root) and a negative one. However, when we use the radical symbol '√', we generally refer to the principal (positive) square root.

• For a number 'x', its square root 'y' satisfies y² = x.
• The symbol for square root is √.
• Example: √16 = 4, because 4² = 16. Also, (-4)² = 16, so -4 is also a square root of 16.

Methods to Find Square Roots

There are several methods to find the square root of a number. For competitive exams, it's crucial to master these methods for speed and accuracy.

Method 1: Prime Factorization Method

This method is best suited for perfect squares. It involves breaking down the number into its prime factors and then pairing them up.

Steps:

1. Find the prime factorization of the given number.
2. Group the identical prime factors in pairs.
3. From each pair, take one factor.
4. Multiply these factors together. The result is the square root of the number.

Example: Find the square root of 576.

Prime factorization of 576: 576 = 2 × 288 = 2 × 2 × 144 = 2 × 2 × 12 × 12 = 2 × 2 × (2 × 2 × 3) × (2 × 2 × 3) = 2 × 2 × 2 × 2 × 3 × 2 × 2 × 3 Rearranging and pairing: 576 = (2 × 2) × (2 × 2) × (2 × 2) × (3 × 3)

Now, take one factor from each pair: √576 = 2 × 2 × 2 × 3 = 24

Method 2: Long Division Method

This method is versatile and can be used to find the square root of any number, whether it's a perfect square or not (giving a decimal approximation). It's particularly useful for larger numbers.

Steps:

1. Group the digits of the number in pairs, starting from the right (for the units place). If the number has an odd number of digits, the leftmost digit remains alone.
2. Find the largest number whose square is less than or equal to the leftmost group (or single digit). This number is the first digit of the square root.
3. Subtract the square of this digit from the leftmost group.
4. Bring down the next pair of digits to form the new dividend.
5. Double the current quotient (the number found so far) and place it on the left as the new divisor's starting part.
6. Find a digit 'x' such that when appended to the doubled quotient and multiplied by 'x', the result is less than or equal to the new dividend. This 'x' is the next digit of the square root.
7. Subtract the product from the new dividend.
8. Repeat steps 4-7 until all pairs have been brought down or until the desired precision is reached.

Example: Find the square root of 625 using the long division method.

1. Group digits: 6 25 2. Leftmost group is 6. The largest square less than or equal to 6 is 4 (2²). So, the first digit of the root is 2. ``` 2 √6 25 4 --- ``` 3. Subtract 4 from 6, which is 2. Bring down the next pair (25) to get 225. ``` 2 √6 25 4 --- 2 25 ``` 4. Double the current quotient (2) to get 4. This is the new divisor's starting part. We need to find a digit 'x' such that 4x * x ≤ 225. Let's try x = 5. Then 45 * 5 = 225. This matches exactly. So, the next digit of the root is 5. ``` 2 5 √6 25 4 --- 45|2 25 2 25 ---- 0 ``` 5. The remainder is 0, so 625 is a perfect square, and its square root is 25.

Method 3: Estimation and Verification (for approximations)

This method is useful for quickly estimating the square root of non-perfect squares or verifying answers.

Steps:

1. Find the two perfect squares that the given number lies between.
2. The square root of the given number will lie between the square roots of these two perfect squares.
3. Estimate the value based on its proximity to the perfect squares.

Example: Estimate √50.

We know that 7² = 49 and 8² = 64. So, 49 < 50 < 64. This means √49 < √50 < √64. Therefore, 7 < √50 < 8. Since 50 is very close to 49, √50 will be slightly greater than 7. A good estimate would be around 7.07.

Square Roots of Decimals

To find the square root of a decimal number, we first pair the digits starting from the decimal point. For the integral part, we pair from right to left. For the fractional part, we pair from left to right. If needed, we can add trailing zeros to make the number of decimal places even.

Example: Find the square root of 12.25.

1. Group digits: 12 . 25 2. Find the largest square less than or equal to 12. It's 9 (3²). The first digit of the root is 3. ``` 3 √12.25 9 --- ``` 3. Subtract 9 from 12, which is 3. Bring down the next pair (25) to get 325. Since we crossed the decimal point, place a decimal point in the quotient. ``` 3. √12.25 9 --- 3 25 ``` 4. Double the current quotient (3) to get 6. Find 'x' such that 6x * x ≤ 325. Try x = 5. 65 * 5 = 325. This matches exactly. The next digit is 5. ``` 3. 5 √12.25 9 --- 65|3 25 3 25 ---- 0 ``` 5. The square root of 12.25 is 3.5.

Properties of Square Roots

Understanding these properties can simplify calculations.

• √ (a × b) = √a × √b
• √ (a / b) = √a / √b
• √a² = |a| (absolute value of a)
• The square root of a negative number is not a real number.
• For any positive number 'x', √x is always positive (principal square root).

Example: Simplify √144.

Using prime factorization: 144 = 2 × 72 = 2 × 2 × 36 = 2 × 2 × 6 × 6 = 2 × 2 × (2 × 3) × (2 × 3) = 2⁴ × 3². √144 = √(2⁴ × 3²) = √(2²) × √(2²) × √(3²) = 2 × 2 × 3 = 12. Alternatively, using the property √a² = a: We know 12² = 144, so √144 = 12.

Example: Simplify √(100/4).

Using the property √ (a / b) = √a / √b: √(100/4) = √100 / √4 = 10 / 2 = 5. Alternatively, 100/4 = 25. √25 = 5.

Solving Problems involving Square Roots

Competitive exams often feature problems that require applying square root concepts in different contexts, such as algebra, geometry, and number theory.

Type 1: Finding the smallest number to multiply or divide to make a number a perfect square.

To find the smallest number to multiply or divide a given number to make it a perfect square, use the prime factorization method.

Steps:

1. Find the prime factorization of the given number.
2. Identify the prime factors that do not have a pair (i.e., appear an odd number of times).
3. To make it a perfect square by multiplication, multiply the number by these unpaired prime factors.
4. To make it a perfect square by division, divide the number by these unpaired prime factors.

Example: Find the smallest number by which 180 must be multiplied to get a perfect square.

Prime factorization of 180: 180 = 18 × 10 = (2 × 9) × (2 × 5) = 2 × 3 × 3 × 2 × 5 = 2² × 3² × 5¹.

The prime factor 5 appears only once (odd number of times). To make it a perfect square, we need another factor of 5. So, the smallest number to multiply by is 5. 180 × 5 = 900. √900 = 30.

Example: Find the smallest number by which 180 must be divided to get a perfect square.

From the prime factorization 2² × 3² × 5¹, the unpaired factor is 5. To make it a perfect square by division, we divide by the unpaired factor. 180 / 5 = 36. √36 = 6.

Type 2: Simplifying expressions with nested square roots.

Expressions like √(a + √b) or √(a - √b) can sometimes be simplified. A common pattern is √(x + y + 2√xy) = √x + √y.

Formula: √(a + 2√b) = √(x) + √(y), where x + y = a and x * y = b. Formula: √(a - 2√b) = √(x) - √(y), where x + y = a and x * y = b.

Example: Simplify √ (7 + 2√10).

Here, a = 7 and b = 10. We need to find two numbers x and y such that x + y = 7 and x * y = 10. The numbers are 5 and 2 (since 5 + 2 = 7 and 5 * 2 = 10). So, √ (7 + 2√10) = √5 + √2.

Example: Simplify √ (11 - 2√28).

Here, a = 11 and b = 28. We need x + y = 11 and x * y = 28. The numbers are 7 and 4 (since 7 + 4 = 11 and 7 * 4 = 28). So, √ (11 - 2√28) = √7 - √4 = √7 - 2. (We take the larger root first to keep the result positive).

Type 3: Problems involving surds (irrational roots).

These problems often involve rationalizing the denominator or simplifying expressions with square roots of non-perfect squares.

Rationalizing the Denominator: To rationalize a denominator containing a square root, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of (a + √b) is (a - √b). The conjugate of (a - √b) is (a + √b). The conjugate of √b is √b.

Example: Rationalize the denominator of 1 / (2 + √3).

Multiply numerator and denominator by the conjugate of (2 + √3), which is (2 - √3): [ 1 / (2 + √3) ] × [ (2 - √3) / (2 - √3) ] = (2 - √3) / [ (2)² - (√3)² ] = (2 - √3) / (4 - 3) = (2 - √3) / 1 = 2 - √3

Example: Simplify √50 + √72 - √18.

First, simplify each term by factoring out perfect squares: √50 = √(25 × 2) = √25 × √2 = 5√2 √72 = √(36 × 2) = √36 × √2 = 6√2 √18 = √(9 × 2) = √9 × √2 = 3√2

Now, substitute these back into the expression: 5√2 + 6√2 - 3√2 Combine the terms as they all have √2: (5 + 6 - 3)√2 = (11 - 3)√2 = 8√2.

Type 4: Square root of large numbers

For very large numbers, the long division method is the most reliable, although time-consuming. However, for numbers that are close to perfect squares or have recognizable patterns, estimation and properties can be very helpful.

Example: Find √1024.

We know 30² = 900 and 40² = 1600. So the root is between 30 and 40. The number ends in 4, so the root must end in 2 or 8. Possible roots are 32 or 38. Let's check 32²: 32 × 32 = 1024. So, √1024 = 32.

Common Mistakes to Avoid

• Confusing square root with square. Remember √9 = 3, not 9.
• Forgetting the negative square root: While the √ symbol usually denotes the principal (positive) root, remember that x² = a has two solutions: √a and -√a.
• Errors in prime factorization or pairing digits in the long division method.
• Incorrectly applying the formula for nested square roots, especially missing the '2' in √(a ± 2√b).
• Calculation errors, particularly with signs or arithmetic.

Practice Problems for RRB Group D

1. Find the square root of 1764. 2. What is the smallest number that must be divided by 1323 to make it a perfect square? 3. Simplify: √ (8 + 2√15). 4. Rationalize the denominator: 1 / (√7 - √5). 5. Calculate √ (0.0529).