If you are looking to answer the following questions: “what is the square root of 513?” The answer is 22.649503305812.
ANSWER: square root of 513=22.649503305812
The square root of a number (513 in this case) is a number (22.649503305812 in this case) which multiplied by itself equals the number from which you are calculating the square root (513). Mathematically square root of 513 can expressed in the radical form or an exponent form as shown below:
- Radical form of the square root of 513: √513
- Exponent form of the square root of 513: (513)½ or (513)0.5
This symbol √ is called radical sign or radix. The number whose square root is being calculated is called the radicand. The radicand is the number underneath the radical sign (radix), in this case it is 513.
Square root of 513 calculator
If you would like to confirm the answer to the question of “what is the square root of 513?” or calculate a square root of any other number use the square root calculator:
Square root of 513 calculation guide
In the following sections we explain how to calculate the square root of 513. We will answer such common questions as: “is the square root of 513 rational or irrational?” and “is 513 a perfect square?” We will discuss what the principal square root of 513 is. For your reference we also included the following tables:
- Nth roots of 513
- Perfect square numbers
- Square roots of numbers around 513
You will find all the information you need on the square root of 513 calculations below.
What is the square root of 513?
The square root of 513 is a number which multiplied by itself equals 513. Therefore, the square root of 513 is 22.649503305812 and we write it √513=22.649503305812 because 22.6495033058122=513. Since 22.6495033058122 is the same as 22.649503305812 x 22.649503305812 the following expressions is true:
513 = 22.6495033058122=22.649503305812×22.649503305812
Therefore
ANSWER: square root of 513=√513=22.649503305812
Is 513 a Perfect Square?
A number is considered a perfect square if it is a product of squaring a whole number (called integer). 0, 1, 2, 3, 4, 7, 8, and 9 are all whole numbers (same as integers). Numbers such as -8, 7.025, or 5 ½ are not whole numbers. Remember squaring is when a number is multiplied by itself. For example, 81 is a perfect square because it is a product of 92 which is the same as 9×9. Notice 9 is a whole number (integer).
In our example the square root of 513 is 22.649503305812. Since 22.649503305812 is not a whole number therefore 513 is not a perfect square.
ANSWER: 513 is not a whole number
Is the square root of 513 rational or irrational number?
A number is rational when it can be made by dividing a whole number by a whole number. A whole number which is also called an integer does not have a fractional part itself. Simply said a number is rational if it can be written as a fraction.
A quick method to determine if 513 is a perfect square. The square roots of perfect squares are all rational numbers. The square roots of numbers that are not a perfect square are all irrational numbers. Since we know that 513 is not a perfect square therefore 513 is not a rational number.
Principal square root of 513
Every positive number has two square roots. One of the square roots is positive and another one is negative. For example the square roots of 513 are -22.649503305812 (negative) and +22.649503305812 (positive) because (-22.649503305812)^2=(+22.649503305812)^2=513. The positive square root is denoted as “the principal square root”. Typically, when we are calculating “the square root of a number” we mean the principal square root. In our case it is positive 22.649503305812 or +22.649503305812 which is written as 22.649503305812. As you may know if there is no minus (-) sign before a number it is a positive number.
Again the principal square root of 513 is 22.649503305812.
Calculating the square root of 513 with a calculator
In order to calculate the square root of 513 with a basic or a scientific calculator you need to enter 513 and then press the key with this sign “√x”
√513 = 22.649503305812
Calculating the square root of 513 in Excel and Google Sheets
There are two functions you can use to do this calculation. You can use the function SQRT () to calculate the square root of any number in Excel and Google Sheets. Click on a cell and type =SQRT(513). Hit enter. You will get 22.649503305812.
You can also use the POWER function which works like an exponent is an equation. To find a square root of 513 type =POWER (513, 1/2). Remember square roots can be expressed as exponents as we discussed earlier in this article.
Table: Nth roots of 513
In the section above we explained how the square root (which is the 2nd root) of 513 is calculated. However there are more roots of 513. Find the Nth roots of 513 in the table below:
| Index (N) | Nth Root of Radicand | Expression | Root |
| 2 | Square Root of 513 | ²√513 | 22.650 |
| 3 | Cube Root of 513 | ³√513 | 8.005 |
| 4 | Forth Root of 513 | ⁴√513 | 4.759 |
| 5 | Fifth Root of 513 | ⁵√513 | 3.484 |
| 6 | Sixth Root of 513 | ⁶√513 | 2.829 |
| 7 | Seventh Root of 513 | ⁷√513 | 2.439 |
| 8 | Eight Root of 513 | ⁸√513 | 2.182 |
| 9 | Nineth Root of 513 | ⁹√513 | 2.000 |
| 10 | Tenth Root of 513 | ¹⁰√513 | 2.000 |
Table: Square root of numbers around 513
This table lists numbers around 513 and their respective square roots.
| NUMBER | SQUARE ROOT |
| 508 | 22.539 |
| 509 | 22.561 |
| 510 | 22.583 |
| 511 | 22.605 |
| 512 | 22.627 |
| 513 | 22.649503305812 |
| 514 | 22.672 |
| 515 | 22.694 |
| 516 | 22.716 |
| 517 | 22.738 |
| 518 | 22.760 |
Table: Perfect square numbers
This table lists perfect square numbers from 1 through 10,000.
| Perfect Squares | Square Root | Whole Number (Integer) |
| 1 | sqrt 1 = | 1 |
| 4 | sqrt 4 = | 2 |
| 9 | sqrt 9 = | 3 |
| 16 | sqrt 16 = | 4 |
| 25 | sqrt 25 = | 5 |
| 36 | sqrt 36 = | 6 |
| 49 | sqrt 49 = | 7 |
| 64 | sqrt 64 = | 8 |
| 81 | sqrt 81 = | 9 |
| 100 | sqrt 100 = | 10 |
| 121 | sqrt 121 = | 11 |
| 144 | sqrt 144 = | 12 |
| 169 | sqrt 169 = | 13 |
| 196 | sqrt 196 = | 14 |
| 225 | sqrt 225 = | 15 |
| 256 | sqrt 256 = | 16 |
| 289 | sqrt 289 = | 17 |
| 324 | sqrt 324 = | 18 |
| 361 | sqrt 361 = | 19 |
| 400 | sqrt 400 = | 20 |
| 441 | sqrt 441 = | 21 |
| 484 | sqrt 484 = | 22 |
| 529 | sqrt 529 = | 23 |
| 576 | sqrt 576 = | 24 |
| 625 | sqrt 625 = | 25 |
| 676 | sqrt 676 = | 26 |
| 729 | sqrt 729 = | 27 |
| 784 | sqrt 784 = | 28 |
| 841 | sqrt 841 = | 29 |
| 900 | sqrt 900 = | 30 |
| 961 | sqrt 961 = | 31 |
| 1024 | sqrt 1024 = | 32 |
| 1089 | sqrt 1089 = | 33 |
| 1156 | sqrt 1156 = | 34 |
| 1225 | sqrt 1225 = | 35 |
| 1296 | sqrt 1296 = | 36 |
| 1369 | sqrt 1369 = | 37 |
| 1444 | sqrt 1444 = | 38 |
| 1521 | sqrt 1521 = | 39 |
| 1600 | sqrt 1600 = | 40 |
| 1681 | sqrt 1681 = | 41 |
| 1764 | sqrt 1764 = | 42 |
| 1849 | sqrt 1849 = | 43 |
| 1936 | sqrt 1936 = | 44 |
| 2025 | sqrt 2025 = | 45 |
| 2116 | sqrt 2116 = | 46 |
| 2209 | sqrt 2209 = | 47 |
| 2304 | sqrt 2304 = | 48 |
| 2401 | sqrt 2401 = | 49 |
| 2500 | sqrt 2500 = | 50 |
| 2601 | sqrt 2601 = | 51 |
| 2704 | sqrt 2704 = | 52 |
| 2809 | sqrt 2809 = | 53 |
| 2916 | sqrt 2916 = | 54 |
| 3025 | sqrt 3025 = | 55 |
| 3136 | sqrt 3136 = | 56 |
| 3249 | sqrt 3249 = | 57 |
| 3364 | sqrt 3364 = | 58 |
| 3481 | sqrt 3481 = | 59 |
| 3600 | sqrt 3600 = | 60 |
| 3721 | sqrt 3721 = | 61 |
| 3844 | sqrt 3844 = | 62 |
| 3969 | sqrt 3969 = | 63 |
| 4096 | sqrt 4096 = | 64 |
| 4225 | sqrt 4225 = | 65 |
| 4356 | sqrt 4356 = | 66 |
| 4489 | sqrt 4489 = | 67 |
| 4624 | sqrt 4624 = | 68 |
| 4761 | sqrt 4761 = | 69 |
| 4900 | sqrt 4900 = | 70 |
| 5041 | sqrt 5041 = | 71 |
| 5184 | sqrt 5184 = | 72 |
| 5329 | sqrt 5329 = | 73 |
| 5476 | sqrt 5476 = | 74 |
| 5625 | sqrt 5625 = | 75 |
| 5776 | sqrt 5776 = | 76 |
| 5929 | sqrt 5929 = | 77 |
| 6084 | sqrt 6084 = | 78 |
| 6241 | sqrt 6241 = | 79 |
| 6400 | sqrt 6400 = | 80 |
| 6561 | sqrt 6561 = | 81 |
| 6724 | sqrt 6724 = | 82 |
| 6889 | sqrt 6889 = | 83 |
| 7056 | sqrt 7056 = | 84 |
| 7225 | sqrt 7225 = | 85 |
| 7396 | sqrt 7396 = | 86 |
| 7569 | sqrt 7569 = | 87 |
| 7744 | sqrt 7744 = | 88 |
| 7921 | sqrt 7921 = | 89 |
| 8100 | sqrt 8100 = | 90 |
| 8281 | sqrt 8281 = | 91 |
| 8464 | sqrt 8464 = | 92 |
| 8649 | sqrt 8649 = | 93 |
| 8836 | sqrt 8836 = | 94 |
| 9025 | sqrt 9025 = | 95 |
| 9216 | sqrt 9216 = | 96 |
| 9409 | sqrt 9409 = | 97 |
| 9604 | sqrt 9604 = | 98 |
| 9801 | sqrt 9801 = | 99 |
| 10000 | sqrt 10000 = | 100 |
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