## Square and Square Roots Questions

Squares and square roots lie at the heart of mathematics, bridging concepts from algebra to geometry. They encompass rational and irrational numbers, where rational numbers can be expressed as fractions and irrational numbers are non-repeating and non-terminating decimals. Understanding squares and square roots is fundamental in solving equations, finding solutions in integer domains, and even in statistical analysis through methods like the least square method, which minimizes the sum of squared differences. Together, they form a cornerstone of mathematical reasoning and problem-solving across various disciplines.

## Solved Problems

**Question.1** : What is the square of 9?

**Answer**: To find the square of a number, you multiply the number by itself. So, the square of 9 is 9×9 = 81. Therefore, 9² = 81.

**Question.2** : Calculate the square root of 64.

The square root of a number is the value that, when multiplied by itself, gives the original number. So, the square root of 64 is a number which, when multiplied by itself, equals 64. In this case, √64 = 8 because 8×8 = 64.

**Question.3** : What is the square of −5?

**Answer**: Similar to finding the square of a positive number, to find the square of a negative number, you multiply the number by itself. So, the square of −5 is (−5)×(−5) = 25. Therefore, (−5)² = 25.

**Question.4** : Find the square root of 144.

**Answer**: Using the same logic as before, the square root of 144 is a number which, when multiplied by itself, equals 144. So, √144 = 12 because 12×12 = 144.

**Question.5** : Calculate the square of 3/4.

A**nswer**: To find the square of a fraction, you square both the numerator and the denominator. So, (3/4)² = (3/4)×(3/4) = 9/16. Therefore, the square of 3/4 is 9/16.

**Question.6** : Explain the concept of perfect squares.

**Answer**: Perfect squares are numbers that result from squaring a whole number. In other words, a perfect square is the square of an integer. For example, 1, 4, 9, 16, 25, 36, and so on are perfect squares because they are the result of squaring integers (1^2, 2^2, 3^2, 4^2, 5^2, 6^2, etc.).

**Question.7** : What is the relationship between squares and square roots?

The square root of a number is the inverse operation of squaring that number. If 𝑥 is the square of 𝑦 (𝑥 = 𝑦²), then the square root of 𝑥 is 𝑦 (√𝑥 = 𝑦). In other words, if 𝑎² = 𝑏, then 𝑎 is the square root of 𝑏, denoted as √𝑏 = 𝑎.

**Question.8**: How do you calculate square roots by hand?

There are various methods for calculating square roots by hand, such as the long division method or the Babylonian method. The long division method involves repeatedly finding quotients and remainders, while the Babylonian method is an iterative algorithm that converges to the square root through successive approximations.

**Question.9** : Explain the properties of squares and square roots.

Some properties of squares and square roots include:

- The square of a negative number is positive: (−𝑎)² = 𝑎².
- The square root of a positive number has two solutions, one positive and one negative.
- The square root of 0 is 0: √0 = 0.
- The square root of a perfect square is an integer.
- Squaring and taking the square root are inverse operations: (√𝑥)² = 𝑥 and (√𝑥²) = ∣𝑥∣.

**Question.10** : How are squares and square roots used in real-life applications?

**Answer**: Squares and square roots have numerous real-life applications, including:

- Calculating areas of squares and rectangles.
- Determining distances in coordinate geometry.
- Estimating in engineering and construction.
- Modeling in physics and engineering.
- Encryption and cryptography in computer science.

**Question.11** : Find the square of −7.

To find the square of a negative number, you simply multiply the number by itself. So, the square of −7 is (−7)×(−7) = 4. Therefore, (−7)² = 49. This result demonstrates that squaring a negative number yields a positive result.

**Question.12** : Determine the square root of 100.

**Answer**: The square root of a number is the value that, when multiplied by itself, equals the original number. So, to find the square root of 100, we’re looking for a number which, when multiplied by itself, gives 100. In this case, √100 = 10 because 10×10 = 100.

**Question.13** : Calculate the square of 5/3.

**Answer**: To find the square of a fraction, you square both the numerator and the denominator separately. So, (5/3)² = (5²/3²) = 25/9. Therefore, the square of 5/3 is 25/9.

**Question.14** : What is the square root of 2?

**Answer**: The square root of 2 is an irrational number, meaning it cannot be expressed as a finite decimal or fraction. It is approximately equal to 1.41421356… The square root of 2 is denoted as √2.

**Question.15** : Explain the concept of negative square roots.

**Answer**: Negative square roots exist because every positive number has two square roots: one positive and one negative. For example, the square root of 9 is both 3 and -3 because 3² = 9 and (−3)² = 9. However, in some contexts, like when dealing with lengths or measurements, we typically consider only the principal (positive) square root.

**Question.16** : Find the square root of 144 and express it in simplified radical form.

The square root of 144 is √144 = 12 because 12×12 = 144. In simplified radical form, this can be written as √144=√12² = 12.

**Question.17** : Determine the square of 1/2.

**Answer**: To find the square of a fraction, you square both the numerator and the denominator separately. So, (1/2)² = (1²/2²) = 1/4. Therefore, the square of 1/2 is 1/4.

**Question.18** : Find the square root of 81.

**Answer**: The square root of 81 is √81 = 9 because 9×9 = 81. It’s important to note that 81 is a perfect square, so its square root is an integer.

**Question.19 **: Calculate the square of −3.

**Answer**: Squaring a negative number yields a positive result. So, the square of −3 is (−3)×(−3) = 9. Therefore, (−3)² = 9.

**Question.20 **: Determine the square root of 49 and express it in simplified radical form.

**Answer**: The square root of 49 is √49 = 7 because 7×7 = 49. In simplified radical form, this can be written as √49 = √7² = 74.

**Question.21 **: Explain the relationship between squares and areas of squares.

**Answer**: The area of a square with side length 𝑠 is given by the formula 𝐴 = 𝑠², where 𝑠*s* is the length of one side of the square. This formula shows that the area of a square is equal to the square of its side length.

**Question.22 **: Find the square root of 16/9.

**Answer**: To find the square root of a fraction, you take the square root of the numerator and the denominator separately. So, √16/9 = √16√9 = 4/3.

**Question.23 **: Determine the square of −4.

**Answer**: Squaring a negative number yields a positive result. So, the square of −4 is (−4)×(−4) = 16. Therefore, (−4)² = 16.

**Question.24 **: Find the square root of 169.

**Answer**: The square root of 169 is √169=13 because 13×13 = 169. Since 169 is a perfect square, its square root is an integer.

**Question.25 **: Calculate the square of 3/5.

**Answer**: To find the square of a fraction, you square both the numerator and the denominator separately. So, (3/5)² = (3²/5²) = 9/25. Therefore, the square of 3/5 is 9/25.

## Square Root Table

Number | Square Root |
---|---|

1 | 1 |

2 | 1.414 |

3 | 1.732 |

4 | 2 |

5 | 2.236 |

6 | 2.449 |

7 | 2.646 |

8 | 2.828 |

9 | 3 |

10 | 3.162 |

11 | 3.317 |

12 | 3.464 |

13 | 3.606 |

14 | 3.742 |

15 | 3.873 |

16 | 4 |

17 | 4.123 |

18 | 4.243 |

19 | 4.359 |

20 | 4.472 |

21 | 4.583 |

22 | 4.69 |

23 | 4.796 |

24 | 4.899 |

25 | 5 |

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## FAQs

**What is a square root, and how does it relate to squares?**

A square root is a value that, when multiplied by itself, equals a given number. For example, the square root of 9 is 3 because 3×3 = 9. Square roots are closely related to squares because finding the square root of a number undoes squaring that number.

**Why are some square roots irrational?**

Some square roots, like the square root of 2 or the square root of non-perfect squares, are irrational because they cannot be expressed as fractions and have non-repeating decimal expansions. These numbers have an infinite number of decimal places without any repeating pattern.

**How are square roots useful in real-life applications?**

Square roots have numerous applications in various fields such as engineering, physics, finance, and computer science. For example, in engineering, square roots are used in calculations involving distances, areas, and volumes. In finance, they’re used in calculating interest rates and investment returns.

**What is the significance of the square root of 1?**

The square root of 1 is 1. In mathematics, this property signifies that any number squared equals 1 has a square root of 1. It’s a fundamental concept that helps understand the inverse relationship between squares and square roots.

**How can I calculate square roots without a calculator?**

There are several methods to calculate square roots manually, such as the long division method, the prime factorization method, or the Newton’s method (also known as the Babylonian method). These methods involve iterative processes to approximate the square root of a given number.

**Why is the square root of 0 equal to 0?**

The square root of 0 is 0 because 0 multiplied by itself equals 0. In other words, any number squared to equal 0 is 0. It’s a fundamental property of square roots and plays a significant role in various mathematical calculations.

**What’s the relationship between the square root table and perfect squares?**

The square root table provides the square roots of numbers from 1 to 25, showcasing the connection between perfect squares (numbers whose square roots are integers) and their respective square roots. For example, the square root of 9 is 3, demonstrating that 9 is a perfect square.

**How do I interpret the decimal values in the square root table?**

The decimal values in the square root table represent the square roots of numbers that are not perfect squares. These decimal values are approximations of the actual square roots and can be used in calculations or measurements where precise values are required.

**Why do we exclude the square root of negative numbers in the table?**

The square root of a negative number is considered to be imaginary and lies outside the scope of real numbers. Therefore, the square root table typically includes only non-negative numbers (0 and positive integers) to provide meaningful and relevant information for real-world applications.

**How can I verify the accuracy of the square root table?**

You can verify the accuracy of the square root table by squaring each number listed and comparing the result to the original number. For example, squaring the square root of 16 should equal 16. This process ensures that the square roots provided in the table are correct and accurate.