# How to Simplify Square Roots Worksheet, Formula, and Definition

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### Key Points about Square Roots

- The square root of a number is a value that, when multiplied by itself, gives the original number.
- Square roots are used in a wide range of mathematical operations, from basic arithmetic to advanced calculus.
- Understanding square roots is crucial for solving equations and real-life problems that involve finding the distance between two points.

## Simplifying Square Roots Easily

The **square root** of a number is a number that when multiplied by itself, gives an original number. **Square roots** are also the inverse operation of **squaring** a number. **Squaring** a number means you multiply that number by itself. The **square root** of a number is the number that gives an **square** when multiplied by itself. You can download our simplify square root worksheet for more practice on how to simplify a **square root**.

Square roots are a fundamental concept in mathematics that are used in a wide range of applications, from basic arithmetic to advanced calculus. Simply put, 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 multiplied by 3 equals 9.

Understanding square roots is crucial for many mathematical operations, such as solving quadratic equations and finding the distance between two points in a coordinate plane. In this article, we will explore the concept of square roots in detail, including how to find the square root of a number, different types of square roots, and examples of square roots in real-life situations. Whether you are a student learning about square roots for the first time or a professional mathematician looking to refresh your knowledge, this article will provide a comprehensive overview of this important mathematical concept.

**Common Core Standard:** 8.EE.A.2**Related Topics:** Cube Roots, Irrational Numbers, Powers of 10, Scientific Notation Intro, Converting Numbers to Scientific Notation, Converting Numbers from Scientific Notation, Adding and Subtracting in Scientific Notation, Multiplying in Scientific Notation, Dividing in Scientific Notation**Return To: **Home, 8th Grade

## What is the Square Root of a Number?

The square root of a number is a value that, when multiplied by itself, gives the original number. It is represented by the radical symbol (√). For example, the square root of 25 is 5, because 5 multiplied by itself equals 25. In mathematical terms, we write it as √25 = 5.

The square root is the inverse operation of squaring a number. Squaring a number means multiplying it by itself. For example, 5 squared is 25 because 5 multiplied by 5 equals 25. Similarly, 7 squared is 49 because 7 multiplied by 7 equals 49.

The square root symbol (√) is used to represent the square root of a number. For example, if we want to find the square root of 16, we write it as √16. The radical symbol is also known as the root symbol or the radical sign.

The square root of a number can be either positive or negative. For example, the square root of 25 is 5, but it can also be -5 because (-5) multiplied by (-5) also equals 25. However, when we talk about the square root of a number, we usually refer to the positive square root.

In summary, the square root of a number is the value that, when multiplied by itself, gives the original number. It is represented by the radical symbol (√) and is the inverse operation of squaring a number. The square root can be either positive or negative, but we usually refer to the positive square root.

## How to find the Square Root of a Number

Finding the square root of a number is the inverse operation of squaring a number. It is the process of determining a number that, when multiplied by itself, results in the original number. The symbol for the square root is √. For example, the square root of 16 is 4 because 4 multiplied by 4 is equal to 16.

### Using a Calculator

One of the easiest and most accurate ways to find the square root of a number is by using a calculator. Most scientific calculators have a square root function that can be accessed by pressing the √ button. Simply enter the value you want to find the square root of and press the √ button to get the result.

### Guess and Check Method

Another way to find the square root of a number is by using the guess and check method. This method involves making an educated guess and then checking to see if the guess is correct. For example, to find the square root of 25, you could start by guessing that the answer is 5. Then, you would check to see if 5 multiplied by itself equals 25. If it does, then your guess was correct. If not, you would need to adjust your guess and try again.

### Value Table Method

The value table method involves creating a table of values to find the square root of a number. To use this method, you would start by listing the perfect squares that are less than the number you want to find the square root of. Then, you would find the difference between the number you want to find the square root of and each perfect square. Finally, you would divide the difference by the square root of the perfect square that is closest to the number you want to find the square root of. The result is an estimate of the square root.

### Long Division Method

The long division method is a more complex way to find the square root of a number. This method involves dividing the number into smaller and smaller parts until the square root is found. The process can be time-consuming and requires a lot of patience and practice.

### Examples

Here are some examples of finding the square root of a number using different methods:

- Using a calculator: The square root of 81 is 9.
- Guess and check method: The square root of 36 is 6.
- Value table method: To find the square root of 20, start by listing the perfect squares that are less than 20: 1, 4, 9, and 16. The difference between 20 and 16 is 4. Divide 4 by the square root of 16, which is 4, to get 1. The estimate of the square root of 20 is 4 + 1 = 5.
- Long division method: To find the square root of 50, start by dividing it into pairs of digits: 5 and 0. Then, find the largest number whose square is less than or equal to 5, which is 2. Write 2 as the first digit of the answer. Subtract 4 from 5 to get 1. Bring down the next pair of digits, which is 00. Double the first digit of the answer (2) to get 4. Find the largest number whose product with 4 is less than or equal to 100, which is 7. Write 7 as the next digit of the answer. Subtract 56 from 100 to get 44. Bring down the next pair of digits, which is 00. Double the answer so far (27) to get 54. Find the largest number whose product with 54 is less than or equal to 4400, which is 8. Write 8 as the next digit of the answer. The final answer is 2.23 (rounded to two decimal places).

In conclusion, there are several methods for finding the square root of a number, including using a calculator, guess and check, value table, and long division. Each method has its own advantages and disadvantages, and the best method to use depends on the situation.

## How to Find Square Root without a Calculator

Finding the square root of a number is an important mathematical operation. While calculators can do the job quickly and easily, it’s also important to know how to find the square root without a calculator. Below are some methods to help you find the square root of a number.

### Long Division Method

One way to find the square root of a number is to use the long division method. This method involves dividing the number into smaller parts until you find the square root. For example, to find the square root of 64, you can divide it into smaller parts like 6 and 4. Then, you can find the square root of each part and combine them to get the final answer.

### Factorisation Method

Another way to find the square root of a number is to use the factorisation method. This method involves finding the factors of the number and pairing them up. For example, to find the square root of 36, you can pair up the factors 2 and 18, and 3 and 12. Then, you can find the square root of each pair and multiply them to get the final answer.

### Division Method

The division method is another way to find the square root of a number. This method involves dividing the number by a series of smaller numbers until you find the square root. For example, to find the square root of 100, you can divide it by 2, then 4, then 5, and so on until you get the answer.

### Visualising

Visualising the square root of a number is another method that can help you find the answer. This method involves picturing a square with an area equal to the number you want to find the square root of. Then, you can divide the square into smaller parts and count them to find the square root.

### Solved Examples

Here are some examples to help you understand how to find the square root of a number:

- Find the square root of 81 using the long division method. Solution: Divide 81 into 8 and 1. The square root of 8 is 2, and the square root of 1 is 1. Therefore, the square root of 81 is 9.
- Find the square root of 49 using the factorisation method. Solution: The factors of 49 are 1, 7, and 49. Pairing up the factors, we get 7 and 7. Therefore, the square root of 49 is 7.

### Practice Questions

Here are some practice questions to help you improve your skills in finding the square root of a number:

- Find the square root of 256 using the division method.
- Find the square root of 144 using the visualisation method.
- Find the square root of 121 using the factorisation method.

By using these methods and practicing regularly, you can become proficient in finding the square root of a number without a calculator.

## 3 Simple Square Roots Examples

Square roots are a fundamental concept in mathematics. They are the inverse operation of squaring a number and are used to find the length of the sides of a square.

- A square root is denoted by a radical symbol.
- When simplifying square roots, you are looking for the number is can be multiplied by itself to get an original number.
- When you find that number, that number is the square root.

Here are some examples of square roots:

- The square root of 9 is 3. This is because 3 multiplied by itself equals 9, making 9 a perfect square.
- The square root of 10 is an irrational number, meaning it cannot be expressed as a simple fraction. It is approximately 3.16227766017.
- The square root of 81 is 9. This is because 9 multiplied by itself equals 81, making 81 a perfect square.
- The square root of 100 is 10. This is because 10 multiplied by itself equals 100, making 100 a perfect square.

It is important to note that not all numbers have a rational square root. For example, the square root of 2 is an irrational number and cannot be expressed as a simple fraction.

Square roots can also be used to solve equations. For example, if an equation is in the form of x^2 = 81, the square root of both sides can be taken to find that x equals 9 or -9.

In summary, square roots are a fundamental concept in mathematics and are used to find the length of the sides of a square. They can also be used to solve equations. While some numbers have rational square roots, others are irrational and cannot be expressed as a simple fraction.

## Square Roots 1 to 100

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

1 | 1 |

2 | 1.414213 |

3 | 1.732051 |

4 | 2 |

5 | 2.236068 |

6 | 2.44949 |

7 | 2.645751 |

8 | 2.828427 |

9 | 3 |

10 | 3.162278 |

11 | 3.316625 |

12 | 3.464102 |

13 | 3.605551 |

14 | 3.741657 |

15 | 3.872983 |

16 | 4 |

17 | 4.123106 |

18 | 4.242641 |

19 | 4.358899 |

20 | 4.472136 |

21 | 4.582576 |

22 | 4.690416 |

23 | 4.795832 |

24 | 4.898979 |

25 | 5 |

26 | 5.09902 |

27 | 5.196152 |

28 | 5.291503 |

29 | 5.385165 |

30 | 5.477226 |

31 | 5.567764 |

32 | 5.656854 |

33 | 5.744563 |

34 | 5.830952 |

35 | 5.91608 |

36 | 6 |

37 | 6.082763 |

38 | 6.164414 |

39 | 6.244998 |

40 | 6.324555 |

41 | 6.403124 |

42 | 6. |

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

43 | 6.557439 |

44 | 6.63325 |

45 | 6.708204 |

46 | 6.78233 |

47 | 6.855655 |

48 | 6.928203 |

49 | 7 |

50 | 7.071068 |

51 | 7.141428 |

52 | 7.211103 |

53 | 7.28011 |

54 | 7.348469 |

55 | 7.416198 |

56 | 7.483315 |

57 | 7.549834 |

58 | 7.615773 |

59 | 7.681146 |

60 | 7.745967 |

61 | 7.81025 |

62 | 7.874008 |

63 | 7.937254 |

64 | 8 |

65 | 8.062258 |

66 | 8.124038 |

67 | 8.185353 |

68 | 8.246211 |

69 | 8.306624 |

70 | 8.3666 |

71 | 8.42615 |

72 | 8.485281 |

73 | 8.544004 |

74 | 8.602325 |

75 | 8.660254 |

76 | 8.717798 |

77 | 8.774964 |

78 | 8.831761 |

79 | 8.888194 |

80 | 8.944272 |

81 | 9 |

82 | 9.055385 |

83 | 9.110434 |

84 | 9.165151 |

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

85 | 9.219544 |

86 | 9.273618 |

87 | 9.327379 |

88 | 9.380832 |

89 | 9.433981 |

90 | 9.486833 |

91 | 9.539392 |

92 | 9.591663 |

93 | 9.643651 |

94 | 9.69536 |

95 | 9.746794 |

96 | 9.797959 |

97 | 9.848858 |

98 | 9.899495 |

99 | 9.949874 |

100 | 10 |

Square root is the inverse operation of squaring a number. It is a value that when multiplied by itself results in the original number. For instance, the square root of 16 is 4 since 4 multiplied by 4 equals 16. In this section, we will look at the square roots of numbers from 1 to 100.

The square roots of numbers from 1 to 100 can be classified into two categories: perfect squares and non-perfect squares. The perfect squares are numbers whose square roots are whole numbers. The non-perfect squares are numbers whose square roots are not whole numbers.

The perfect squares from 1 to 100 are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. The square roots of these numbers are integers. For example, the square root of 9 is 3, and the square root of 64 is 8.

On the other hand, the non-perfect squares from 1 to 100 are numbers whose square roots are not integers. These numbers have decimal representations that go on infinitely without repeating. For instance, the square root of 2 is approximately 1.41421356, and the square root of 7 is approximately 2.64575131.

It is worth noting that the square roots of even perfect squares are even integers, while the square roots of odd perfect squares are odd integers. For example, the square root of 16 is 4, which is even, while the square root of 25 is 5, which is odd.

In conclusion, the square roots of numbers from 1 to 100 can be classified into perfect squares and non-perfect squares. The perfect squares have integer square roots, while the non-perfect squares have non-integer square roots. Additionally, even perfect squares have even integer square roots, while odd perfect squares have odd integer square roots.

## Square Roots Definition

A square root of a number is a value that, when multiplied by itself, gives the number. In other words, it is a factor of a number that when squared gives the number. For example, the square root of 9 is ±3 because 3 x 3 = 9 and (-3) x (-3) = 9.

Square roots can be positive or negative, depending on whether the number being rooted is positive or negative. For instance, the square root of 16 is 4 or -4, because 4 x 4 = 16 and (-4) x (-4) = 16. On the other hand, the square root of -16 is not a real number, since no real number multiplied by itself can equal a negative number.

It is important to note that the symbol √ always means the positive square root. For instance, √16 = 4, not -4. However, when dealing with equations, both positive and negative square roots must be considered.

In mathematics, the term “nonnegative” is used to describe numbers that are greater than or equal to zero. The square root of a nonnegative number is always a nonnegative number. For example, the square root of 0 is 0, and the square root of 25 is 5.

In summary, a square root is a value that when multiplied by itself produces the original number. It can be positive or negative, depending on whether the number being rooted is positive or negative. The symbol √ always means the positive square root, but when dealing with equations, both positive and negative square roots must be considered. Finally, the square root of a nonnegative number is always a nonnegative number.

## Square Roots Formula

The square root of a number is a value that when multiplied by itself gives the original number. The number under the radical sign is called the radicand.

The formula to calculate the square root of a number is:

```
√x = y
```

where `x`

is the radicand and `y`

is the square root.

For example, the square root of 25 is 5, because 5 times 5 equals 25. In this case, `x`

is 25 and `y`

is 5.

To multiply square roots, you can use the following formula:

```
√x * √y = √(x * y)
```

For example, to multiply the square root of 4 and the square root of 9, you can use the formula:

```
√4 * √9 = √(4 * 9) = √36 = 6
```

In this case, the radicand of the first square root is 4, and the radicand of the second square root is 9. Multiplying them together gives 36, which has a square root of 6.

It is important to note that the square root of a negative number is not a real number. However, it is possible to work with imaginary numbers, which are denoted by the symbol `i`

.

Overall, the square roots formula is a basic concept in mathematics that is used to solve many different types of problems. By understanding this formula and how to use it, individuals can gain a deeper understanding of mathematical concepts and improve their problem-solving abilities.

## Square Roots in Equations

Square roots can appear in various equations, including quadratic equations, calculus problems, and equations involving exponents. When solving equations with square roots, it is important to keep in mind that the principal root is the non-negative root.

In quadratic equations, the square root may appear when completing the square or when using the quadratic formula. For example, consider the equation x^2 + 6x + 5 = 0. Completing the square yields (x + 3)^2 – 4 = 0, which can be simplified to (x + 3)^2 = 4. Taking the square root of both sides gives x + 3 = +/- 2, and solving for x yields x = -1 or x = -5.

In calculus, square roots can appear when finding the derivative or integral of a function. For example, when finding the derivative of f(x) = sqrt(x), the chain rule is used to obtain f'(x) = 1/(2sqrt(x)).

Equations involving exponents may also include square roots. For example, consider the equation 2^(x/2) = 8. Taking the logarithm of both sides yields (x/2)log(2) = log(8), which simplifies to x = 6.

It is important to note that not all square root equations have solutions. For example, the equation sqrt(x) = -1 has no real solutions. When working with square roots in equations, it is important to check for extraneous solutions by plugging in the solution and verifying that it satisfies the original equation.

Overall, square roots can appear in various equations and it is important to understand how to solve them. By keeping in mind the principal root and checking for extraneous solutions, one can confidently solve equations involving square roots.

## Types of Square Roots

Square roots can be classified into three types – rational, irrational, and complex.

### Rational Square Roots

A rational square root is a square root of a perfect square, which is a number that can be expressed as the product of two equal integers. For example, the square root of 4 is 2, and the square root of 9 is 3. Rational square roots are always positive and can be expressed as a fraction.

### Irrational Square Roots

An irrational square root is a square root of a number that is not a perfect square. Irrational square roots cannot be expressed as a fraction and have an infinite number of non-repeating decimals. Examples of irrational square roots include the square root of 2, the square root of 3, and the square root of 5.

### Complex Square Roots

A complex square root is a square root of a negative number. Complex square roots are expressed in terms of the imaginary unit, i, which is defined as the square root of -1. For example, the square root of -4 is 2i, and the square root of -9 is 3i. Complex square roots always come in pairs, with one positive and one negative.

In summary, square roots can be classified into rational, irrational, and complex types. Rational square roots are square roots of perfect squares and can be expressed as a fraction. Irrational square roots are square roots of non-perfect squares and cannot be expressed as a fraction. Complex square roots are square roots of negative numbers and are expressed in terms of the imaginary unit.

## How to do Square Roots FAQ

### How do you simplify square roots?

To simplify a square root, you need to factorize the number inside the radical sign into its prime factors. Then, you can take out any pairs of identical factors and write them as a single factor outside the radical sign. For example, the square root of 36 can be simplified as the square root of 6 times 6, which is 6.

### What is the difference between a perfect square and a square root?

A perfect square is a number that has an integer square root. For example, 9 is a perfect square because its square root is 3, which is an integer. On the other hand, a square root is the inverse operation of squaring a number. For example, the square root of 16 is 4, because 4 times 4 equals 16.

### How do you find the square root of a decimal?

To find the square root of a decimal, you can use a calculator or estimate the answer by finding the two perfect squares that the decimal is between. Then, you can use the fact that the square root of any number between two perfect squares is between the square roots of those two perfect squares.

### What is the Pythagorean theorem and how is it related to square roots?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is related to square roots because to find the length of the hypotenuse, you need to take the square root of the sum of the squares of the other two sides.

### How do you solve equations involving square roots?

To solve an equation involving square roots, you need to isolate the square root on one side of the equation and square both sides. However, you need to be careful when doing this because squaring both sides of an equation can introduce extraneous solutions.

### What are some real-world applications of square roots?

Square roots have many real-world applications, such as in construction, engineering, and physics. For example, calculating the distance between two points in three-dimensional space involves taking the square root of the sum of the squares of the differences between the coordinates of the two points.

## Squares and Square Roots Worksheet Video Explanation

Watch our free video on **how to find Square Roots**. This video shows how to solve problems that are on our free **How to Simplify Square Roots **worksheet that you can get by submitting your email above.

**Watch the free Simplifying Square Roots video on YouTube here: How to Simplify Square Roots**

**Video Transcript:**

This video is about how to simplify square roots. You can get the worksheet we use in this video for free by clicking on the link in the description below.

We’re talking about how to simplify square roots. You have to know what a square root is. A square root is another way of saying what number, times itself, takes us back to an original number. Another way of thinking about this is to use actual numbers. This will show you how to solve square roots. You can download our simplifying square roots worksheets for more practice.

If we wanted to find the square root of, let’s say 9. what we would need to find is what number times itself would get us back to our original number, which in this case was 9. There is only one number that when you multiply times itself will get us back to 9, in this case the number is 3. The square root of 9 is 3 because 3 times 3 equals 9.

Another important thing to mention is this symbol. This symbol denotes when you are finding the square root of a number. The symbol is called the radical symbol. Anytime you see the radical sign with a number under it what that means is you are looking for the square root of that number. If you see the radical sign and then 16 under it, what that’s saying is you need to find the square root of 16, or in other words you need to find the number that when multiplied times itself gets us back to 16. In this case it would be 4 because 4 times 4 equals 16.

The last thing I want to mention is something called perfect squares. A perfect square is when you take the square root of a number and the root is a whole number. The square root of 100 for example is a perfect square because the answer is 10. However the square root of 56 is not a perfect square because it’s answer is 7.48. The square root of 100 is 10 which is a whole number, which makes it a perfect square and then the square root of 56 is 7.48 which is not a whole number and is not a perfect square.

Let’s jump down to a couple example problems to show you how to simplify square roots. When simplifying number four it says find the square root of 64. What you are looking for is what number times itself will get us back to 64. In this case the square root of 64 would be 8 because 8 times 8 equals 64. Our answer is going to be 8. Number five gives us a square root of 16. You have to think what number times itself gives us 16. 4 times 4 is 16 so the square root of 16 is equal to 4.

The last two examples of how the simplify square roots were going to show you is number 9 and number 10. Number 9 says find the square root of 36. Well the square root of 36 is going to be 6 because 6 times 6 is 36. Number 10 says find the square root of 4. The square root of 4 has to be 2 because 2 times 2 is 4.

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