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Math Calculators ▶ Square Root Calculator

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An online square root calculator helps you to find the square & nth root of any positive as well as negative numbers. Also, this sqrt calculator shows you whether the given number is a perfect square root or not. For example; \(4\), \(9\) & \(16\) are the perfect squares of \(2\), \(3\) & \(4\) respectively.

This context is packed with lots of helpful information including the square root formula, how to find it sqrt step-by-step & much more. Read on!

You can check an online factor calculator if you need to determine the factors and the pairs of factors of positive or negative integers.

In mathematical terms, a square root of a number ‘\(x\)’ is referred to as a number ‘\(y\)’ such that \(y^2 = \text { x}\); in other words, a factor of a number that, when multiplied by itself equal to the original number.

For example, \(3\) and \(-3\) are said to be as the square roots of 9, since \(3^2 = (-3) ^2 = 9\). You can try the square root calculator to simplify the principal square root for the given input.

The given formula is considered to represent the square root:

$$ \sqrt[n]{x} = x^\frac {1}{n} $$

Remember that every nonnegative real number ‘x’ has a unique nonnegative square root that is known as principal square root, it is denoted by \(\sqrt{x}\), where \(\sqrt{ } \) symbol called the radical sign or radix. For example, the principal square root of \(49\) is \(7\), which denoted \(\sqrt{49} = 7\). However, the number whose square root is being considered is said to be the radicand. So, you can check the complex or imaginary solutions for square roots of negative real numbers by using square roots calculator.

Also, you can try our online exponent calculator that helps you to calculate the value of any number raised to any power.

To prepare for the calculation of square root, then you should remember the basic perfect square root. As the sqrt of \(1, 4, 9, 16, 25, 100\) is \(1, 2, 3, 4, 5,\) and \(10\).

To find the sqrt of \(\sqrt{25}\), let’s see!

\(\sqrt {25} = \sqrt{5 \times 5}\)

\(\sqrt {25} = \sqrt{5^2}\)

\(\sqrt {25} = 5\)

These are the simplest square roots because they give every time an integer, but what when a number has not a perfect square root? For example, you have to estimate the sqrt of 54?

- As you know \(sqrt {49} = 7\) & \(\sqrt {64} = 8\). So, the \(\sqrt {54}\) is between the \(8\) and \(7\).
- The number \(54\) is closer to \(49\) than \(64\). So, you can try guessing \(\sqrt {54} = 7.45\)
- Then, by squaring \(7.45\), \((7.45)^2 = 55.5\) which is greater than \(54\). So you should try the smaller number. Let’s take \(7.3\)
- By taking the square of \(7.3\), it gives \(53.29\) which is close to \(54\).
- It means the square root of \(54\) is between \(7.3\) & \(7.4\).

Let’s take another example:

**Example:**

What is a square root of \(27\)?

**Solution:**

As the \(27\) is not the perfect square of any number. So, we have to simplify it as:

\(\sqrt {27} = \sqrt {9 \times 3}\)

\(\sqrt {9} \times \sqrt {3} = \sqrt[3] {3}\)

Our square root calculator considers these formulas & simplification techniques to solve the sqrt of any number or any fraction.

The sqrt of fractions can be determined by the division operation. Look at the following example:

$$ (\frac {a}{b})^{\frac {1}{2}} = \frac {\sqrt {a}}{\sqrt{b}} = \sqrt{\frac {a}{b}} $$

Where \(\frac {a}{b}\) is any fraction. Let’s have another example:

**Example:**

What is square root of \(\frac {9}{25}\)?

**Solution:**

\(\sqrt{\frac {9}{25}} = \frac {\sqrt {9}}{\sqrt {25}}\)

\(\frac {\sqrt {9}}{\sqrt {25}} = \frac {3}{5} = 0.6\)

√9 / √25 = 3 / 5 = 0.6

At school level, we have been taught that the square root of negative numbers cannot exist. But, the mathematicians introduce the general set of numbers (Complex numbers). As:

$$ x = a + bi $$

Where, a is real number & b is an imaginary part. The iota \((i)\) is a complex number with a value:

\(i = \sqrt {-1}\) . Let’s have some examples:

The sqrt of \(-4\) = \(\sqrt {-4} = \sqrt {-1 \times 9} = \sqrt {(-1)} \sqrt {9} = 3i\)

What is the square root of -17 = √-17 = √-1 * 17 = √ (-1) √17 = 17i

There’s no doubt, finding square root manually quite complex, but becomes easy with this roots calculator. Hold the given steps and get the exact sqrt calculations:

Read on!

**Inputs:**

- First of all, hit the tab to choose the square root or nth root for any number.
- Very next, enter the number for which you want to do the calculation according to the selected option.
- Lastly, click on calculate button.

**Outputs:**

Once done, the calculator shows:

- Square root of the number.
- The nth root of the number.
- Step-by-Step calculation
- Tells either the number has a perfect square root or not

**Note:**

It doesn’t matter at all what values you entered, this online square roots calculator simplifying the given number accurately by using the sqrt maths formula.

Yes, the positive numbers have more than one sqrt, one is positive & the other is negative.

No, it is an irrational number.

Reason:

The square root of 2 cannot be expressed as the quotient of two numbers.

Some roots are rational while others are irrational.

**Step 1:**Estimate: first of all, estimate the square root. You need to get as close as you can by simply determining two perfect square roots the given number is between**Step 2:**Divide: now, you need to divide the given number by one of those square roots**Step 3:**Average: you need to take the average of the result of step 2 as well as the root**Step 4:**Now, you have to use the result of step 3 to repeat steps 2 & 3 until you get a number, which is accurate enough for the solution.

To solve an equation that has a square root in it:

- First, you need to isolate the square root on one side of the given equation
- Then, simply square both sides of the equation and keep solving for the variable
- Finally, verify your work, all you need to substitute the obtained value of variable into the original equation

x | √x |
---|---|

1 | 1 |

2 | 1.41421 |

3 | 1.73205 |

4 | 2 |

5 | 2.23607 |

6 | 2.44949 |

7 | 2.64575 |

8 | 2.82843 |

9 | 3 |

10 | 3.16228 |

11 | 3.31662 |

12 | 3.4641 |

13 | 3.60555 |

14 | 3.74166 |

15 | 3.87298 |

16 | 4 |

17 | 4.12311 |

18 | 4.24264 |

19 | 4.3589 |

20 | 4.47214 |

-15 | 3.87i |

-18 | 4.24i |

-32 | 5.66i |

-49 | 7i |

-64 | 8i |

73 | 8.54 |

-81 | 9i |

194 | 13.93 |

200 | 14.14 |

205 | 14.32 |

208 | 14.42 |

325 | 18.03 |

360 | 18.97 |

1500 | 38.73 |

2000 | 44.72 |

2100 | 45.83 |

Find the square root of… | The square root | |
---|---|---|

^{2}√ 1 |
1.0000000000 | |

^{2}√ 2 |
1.4142135624 | |

^{2}√ 3 |
1.7320508076 | |

^{2}√ 4 |
2.0000000000 | |

^{2}√ 5 |
2.2360679775 | |

^{2}√ 6 |
2.4494897428 | |

^{2}√ 7 |
2.6457513111 | |

^{2}√ 8 |
2.8284271247 | |

^{2}√ 9 |
3.0000000000 | |

^{2}√ 10 |
3.1622776602 |

Square roots are frequently appearing in mathematical formulas including quadratic formulas, discriminant as well as in many physics laws. Further, it is used in many places in daily life, used by engineers, carpenters construction managers, medical assistants, and many others. When it comes to calculations for a large number, it is very tricky & complex. Simply, try the online square root calculator that helps you to determine the square root according to your need.

From the source of Wikipedia : Definition, uses & properties

From the site of virtualnerd : Square root of fractions

From the source of khanacademy : Square root of negative numbers

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