Square root is the reverse operation of squaring the number i.e. for square the number we use \( x^2 \) and for square root the number we use \( x^{1/2} \). Example: \( \sqrt{36} = 6 \) (because \( 6 × 6 = 36 \), so the square root of 36 is 6. Note that (−6) × (-6) = 36 too, so -9 is also a square root of 36. The radical (i.e. √) which always means the positive square root.

# Square Root Calculator

### Table of Contents

- What is the meaning of the symbol of the square root?
- Principal square root
- Rational numbers square root
- Square root of irrational numbers
- Principal square root of non-negative number
- Significance of signs in square root
- Estimating the Square Root of Numbers to the Nearest Hundred
- Uses of square root in practical life (Hero’s formula and Pythagoras theorem)
- Find out the length of articles by applying Pythagoras theorem
- How to simplify square root?
- FAQs

## What is the meaning of the symbol of the square root?

The meaning of the square root symbol \(√\) is a positive number whose square root is x.

Example:

\( \sqrt{9} = 3\)

\( \sqrt{9} = \pm 3\)

\( \pm \sqrt{9} = \pm 3\)

\( -\sqrt{9} = -3\)

\( f(x) = \sqrt{x}\)

\(x^2 = 9\)

\(\pm \sqrt{x^2} = \pm \sqrt{9}\)

\(x = \pm 3\)

Let \(\sqrt{2378} \)

max \((\sqrt{2378})\)

\(+\sqrt{2378} \) and \(-\sqrt{2378} \) both roots

## Principal square root

Principal square root is a positive square root of a number it can be rational or irrational.

## Rational numbers square root

Let’s take \(3^2\) and \((-3)^2\).

For \(3^2\), \(3^2 = 3 ✕ 3 = 9 = \sqrt{9}\)

For \((-3^)2\), \((-3)^2 = (-3) ✕ (-3) = 9 = \sqrt{9}\)

Let’s take \(4^2\) and \((-4)^2\).

For \(4^2\), \(4^2 = 4 ✕ 4 = 16 = \sqrt{16}\)

For \((-4^)2\), \((-4)^2 = (-4) ✕ (-4) = 16 = \sqrt{16}\)

## Perfect Squares

Product of multiplying equal numbers. Examples of perfect square:

1- \( \sqrt{64} = 8 \)

2- \( \sqrt{121} = 11 \)

3- \( \sqrt{0.04} = 0.2 \)

4- \( \sqrt{0.49} = 0.7 \)

5- \( \sqrt{\frac{9}{25}} = \frac{3}{5} \)

## Square Root of Irrational Numbers

Irrational numbers are,

1- Non perfect square numbers

2- Non repeating

3- Non terminating decimals

**Example:**

\( \sqrt{2} = 1.414…. \)

\( \sqrt{5} = 2.236…. \)

## Principal Square Root of Non-Negative Number

\( x^2 = 4 \)

\( x = 2 => 2^2 = 4 \)

\( x = -2 => (-2)^2 = 4 \)

\( \sqrt{4} = 2\) or \( -2 \) This notation refers specifically to the principal square root of 4 so, principal square root of non-negative number is non negative therefore \( \sqrt{4} = 2 \).

## Significance of Signs in Square Root

\( x^2 = 4 \)

\( x = 2 \)

Here one solution is missing.

\( x^2 = 4 \)

\( x^2 – 4 = 0 \)

\( x = 2 \) and \( x = -2 \)

\( 2^2 = 4 \) and \( (-2)^2 = 4 \)

## Estimating the Square Root of Numbers to the Nearest Hundred

\( \sqrt{56} \)

Sort out the perfect square root between above and below of 56 those are 7 and 8.

\( (7.5)^2 = 56.25 \)

\( (7.4)^2 = 54.76 \)

\( (7.48)^2 = 55.95 \)

So, \( \sqrt{56} = 7.48 \)

## Estimating the square root of 5 to the nearest hundred

**Step 1:** Find the two consecutive perfect squares which \(\sqrt{5}\) lies.

So, square root of \(\sqrt{5}\) lies between 2 and 3.

**Step 2:** Estimate the square root to the nearest hundred by using number line.

*NOTE: If the number is closer to the lower perfect square, adding 0.10 – 0.40 to the whole number and if the number is closer to the higher perfect square adding 0.50 – 0.90 to the whole number.*

## Alternate method for estimating the \(\sqrt{5}\) to the nearest hundred

X | X^2 | Target 5 |
---|---|---|

2.10 | 4.41 | Closer to 5 |

2.20 | 4.84 | Closer to 5 |

2.30 | 5.29 | Exceed to 5 |

2.21 | 4.8841 | Closer to 5 |

2.22 | 4.92 | Very close to 5 |

2.23 | 4.97 | Very close to 5 |

2.24 | 5.0176 | Exceed to 5 |

2.236 | 4.996 | Very close to 5 |

From 2.31 to 2.235 it is still very close to 5 but 2.236 is the closest and we need to round it to the nearest hundred so, the closest nearest estimated value is 2.24.

Hence, \(\sqrt{5} = 2.24\)

## Uses of square root in practical life (Hero’s formula and Pythagoras theorem)

For calculating area of a specific figure by using hero’s formula:

\(area = \sqrt{s(s – a)(s – b)(s – c)}\)

Here “s” is semi perimeter.

\( S = \dfrac{a + b + c}{2} \)

**Example:**

Consider \( a = 11 \), \( b = 17 \) and \( c = 16 \).

\( S = \dfrac{a + b + c}{2} \)

\( S = \dfrac{11 + 17 + 16}{2} \)

\( S = 22 \)

Applying Hero’s formula for calculating area

\(area = \sqrt{s(s – a)(s – b)(s – c)}\)

\(area = \sqrt{22(22 – 11)(22 – 16)(22 – 17)}\)

\(area = \sqrt{7620}\)

\(area = 85.5206 \) sq units

## Find out the length of articles by applying Pythagoras theorem.

**A 32 inch TV is to be bought with dimension 16 and 9.How actually big would be the TV length wise and breadth wise?**

Let the common multiple to be x

\(a = 16x\) and \(b = 9x\)

By applying Pythagoras Theorem

\(C^2 = A^2 + B^2\)

\(32^2 = (16x)^2 + (9x)^2\)

\(x = 1.743 \) inch

So, length and breadth of TV is

\(a = 16x = 16(1.74) = 27.838 \) inch

\(b = 9x = 9(1.74) = 15.687\) inch

## How to simplify square root?

1- \(\sqrt{49} = 7\)

2- \(\sqrt{-25} = \sqrt{25} \sqrt{-1} = 5i\)

3- \(-\sqrt{81} = -(9) = -9\)

4- \(-\sqrt{-64} = -\sqrt{64} \sqrt{-1} = -(8)i = -8i\)

5- \(\sqrt{18} = \sqrt{9} \sqrt{2} = 3\sqrt{2}\)

6- \(\sqrt{75} = \sqrt{25} \sqrt{3} = 5\sqrt{3}\)

7- \(4\sqrt{98} = 4\sqrt{49} \sqrt{2} = 4(7)\sqrt{2} = 28\sqrt{2}\)

## FAQS

**Q1: Why is it called as square root?**

Answer: Because it is the root of the square multiplying a number by itself gives you a square.

**Q2: What is not allowed in a square root?**

Answer: Negative number don’t have real square root as a square is either positive or zero. The square roots of a number that are not a perfect square are member of the irrational numbers.

**Q3: Can we separate numbers (in addition subtraction form) under a square root?**

Answer: We cannot split a sum or difference as it is a unacceptable. (Violation of the rule).

\(\sqrt{x + y} ≠ \sqrt{x} + \sqrt{y}\)

**Q4: Why is square root important?**

Answer: Because they show up, when we compute areas, which is very much required in practical life.

**Q5: Why do we need square root in real life?**

Answer: Square root has many applications from Hero’s formula to the Pythagoras theorem for finding the area when the lengths are given or we can say that sides are shared. It is very useful in architecture, engineering work and the construction of anything, especially bridges construction and last but not least in carpentering work.

**Q6: What the symbol of square root?**

Answer: Radical, the √ is a symbol of square root, normally used to denote square root or the nth root.

**Q7: How does square root work?**

Answer: There are multiple approaches here we shared square root by prime factorization method.

Step 1: Divide the given number into its prime factors.

Step 2: Form pairs of simple factors such that both factors in each one are equal.

Step 3: Take one factor from the pair.

Step 4: Find the product of the factors obtained by taking one factor from each pair.

Example:

2 | 324 |

2 | 162 |

3 | 81 |

3 | 27 |

3 | 9 |

3 | 3 |

1 |

\(2 ✕ 2 ✕ 3 ✕ 3 ✕ 3 ✕ 3\)

\(2 ✕ 3 ✕ 3 = 18\)

Hence, \(\sqrt{324} = 18\)

**Q8: What is a hidden fact of the square root?**

Answer: Only number bigger than or equal in zero have real square roots.

**Q9: Which number is the square of itself?**

Answer: Only 1, as it is the only natural number that is the square of itself.

**Q10: Is 4000 square number?**

Answer: 4000 is not a perfect square so; it is not a square number.

**Q11: What is the largest perfect square for 28?**

Answer: If we are referring to a perfect square factors that is a factor of 28, the only one is 4.

**Q12: Is 91 a perfect square?**

Answer: The prime factorization of 91 is \(7 ✕ 13\) so the prime factor 7 is not in pair so it is not a perfect square.

**Q13: What are the first 5 square numbers?**

Answer: First five whole square numbers are 0,1,2,4 and their squares are 0,1,4,9 and 16.

**Q14: How to calculate non perfect square root?**

Answer: Let’s take example \sqrt{7} = ?

By using a quick trick,

\(\sqrt{x \pm y} = \sqrt{x} \pm \frac{y}{2\sqrt{x}} => \sqrt{9 – 2} = 3 – \frac{2}{2(3)} => 3 – \frac{2}{6} => 3 – \frac{1}{3} => \frac{9 – 1}{3} => \frac{8}{3}\)

Hence, \(\sqrt{7}\) square root is 2.66.

**Q15: What is the largest square root of -8?**

Answer: If we are referring to a perfect square factors that is a factor of 8, the only one is 2.

**Q16: What is the square root of 4?**

Answer: Square root of 4 is 2.

**Q17: What is the square root of 7?**

Answer: Square root of 7 is 2.6457513110646

**Q18: What is the square root of 400?**

Answer: Square root of 400 is 20