square root of 85

What does square root of 85 mean?

The square root of 85 is a mathematical operation that determines a number which, when multiplied by itself, equals 85. In other words, it finds the value that, when squared, gives the result of 85.

What is the answer to square root of 85?

The answer to the square root of 85 is approximately 9.21954445729.

Is square root of 85 a rational number or an irrational number?

The square root of 85 is an irrational number. An irrational number is a number that cannot be expressed as a fraction or a ratio of two integers. The decimal representation of the square root of 85 goes on indefinitely without repeating or terminating, indicating its irrationality.

Solving methods for square root of 85

There are several methods to solve for the square root of 85. Here, we will use the long division method to find an approximation.

Step-by-step solution:

1. Start by assuming a number, let's say 9, as the first digit of the square root.
2. Multiply this assumed digit by itself: 9 * 9 = 81.
3. Subtract the result from 85: 85 - 81 = 4.
4. Bring down the next two digits of 85, which is 00.
5. Double the assumed digit and write it as the divisor: 9 * 2 = 18.
6. Find the largest digit to fill in the blank such that the product of the divisor and the blank, when multiplied by the blank, is less than or equal to 400. In this case, the largest digit is 2.
7. Multiply the divisor by the blank and write the result below the 400: 18 * 2 = 36.
8. Subtract the result from 400: 400 - 36 = 364.
9. Bring down the next two digits of 85, which is 00.
10. Double the assumed digit and append a blank to it: 92.
11. Find the largest digit to fill in the blank such that the product of the divisor and the blank, when multiplied by the blank, is less than or equal to 364. In this case, the largest digit is 3.
12. Multiply the divisor by the blank and write the result below the 364: 92 * 3 = 276.
13. Subtract the result from 364: 364 - 276 = 88.
14. Bring down the next two digits of 85, which is 00.
15. Double the assumed digit and append a blank to it: 923.
16. Find the largest digit to fill in the blank such that the product of the divisor and the blank, when multiplied by the blank, is less than or equal to 880. In this case, the largest digit is 9.
17. Multiply the divisor by the blank and write the result below the 880: 923 * 9 = 8307.
18. Subtract the result from 880: 880 - 8307 = -7427.

At this point, we have obtained a negative result, indicating that the approximation is sufficient. Therefore, the square root of 85 is approximately 9.21954445729.

More than 3 solved examples about square root of 85 and detailed explanation

Example 1: To find the square root of 85, we can use the long division method as explained above. The approximation is 9.21954445729.

Example 2: Another way to approximate the square root of 85 is by using a calculator. By entering the value 85 and pressing the square root button, the calculator will give us the result of approximately 9.21954445729.

Example 3: We can also use the Newton-Raphson method to find the square root of 85. By starting with an initial guess, such as 10, and iteratively applying the formula x = (x + 85/x) / 2, we can converge to the square root of 85. After a few iterations, we will obtain the approximation of approximately 9.21954445729.

What does square root mean in mathematics?

In mathematics, the square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the symbol √ and is used to find the side length of a square with a given area or to solve quadratic equations. The square root is the inverse operation of squaring a number.

Square root formula

The square root of a number x can be calculated using the formula:

√x = x^(1/2)

This formula raises x to the power of 1/2, which is equivalent to taking the square root.

What symbols are used to represent square root?

The symbol √ is commonly used to represent the square root in mathematics. It is placed in front of the number or expression for which the square root is being calculated.

What are the types of square roots?

There are two types of square roots: positive and negative. The positive square root is the principal square root and is denoted by √x. It represents the positive value that, when squared, gives the original number. The negative square root is denoted by -√x and represents the negative value that, when squared, gives the original number. In most cases, when we refer to the square root of a number, we are referring to the positive square root.

FAQs on square root of 85

Q: What is the square root of 85? A: The square root of 85 is approximately 9.21954445729.

Q: Is the square root of 85 a rational number? A: No, the square root of 85 is an irrational number.

Q: Can the square root of 85 be simplified? A: No, the square root of 85 cannot be simplified further as it is already in its simplest form.

Q: How can I calculate the square root of 85 without a calculator? A: You can use the long division method or the Newton-Raphson method to approximate the square root of 85 manually.

Q: What is the square root of a negative number? A: The square root of a negative number is not a real number. It is represented by the imaginary unit "i" and is used in complex numbers.