Quick Answer

The square root of 144 is exactly 12 (a perfect square). The square root of 50 is approximately 7.071, which simplifies to 5 times the square root of 2.

Number

Common Examples

Input	Result
144	12 (perfect square)
50	7.0711 (simplified: 5*sqrt(2))
2	1.4142 (simplified: sqrt(2))
200	14.1421 (simplified: 10*sqrt(2))
81	9 (perfect square)

How It Works

The square root of a number n is defined as the value x where:

x = sqrt(n), meaning x * x = n

Square roots fall into two categories:

• Perfect squares have integer square roots. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
• Non-perfect squares have irrational square roots that can often be simplified using radical notation.

Simplifying Radicals

To simplify a square root, find the largest perfect square factor of the number under the radical. Then apply the property:

sqrt(a * b) = sqrt(a) * sqrt(b)

For example, to simplify sqrt(72):

1. Find the largest perfect square factor of 72. Since 72 = 36 * 2, and 36 is a perfect square, use that.
2. sqrt(72) = sqrt(36 * 2) = sqrt(36) * sqrt(2) = 6 * sqrt(2).

Worked Example

To find the square root of 50: The largest perfect square factor of 50 is 25, since 50 = 25 * 2. Therefore sqrt(50) = sqrt(25) * sqrt(2) = 5 * sqrt(2). As a decimal, sqrt(50) = 7.07107 (approximately). Since 50 is not a perfect square, the result is irrational and the decimal continues without repeating.

For a perfect square like 144: sqrt(144) = 12, because 12 * 12 = 144. No further simplification is needed.

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Frequently Asked Questions

What is a perfect square?

A perfect square is a number that has an integer as its square root. For example, 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are the first ten perfect squares. The number 144 is also a perfect square because 12 x 12 = 144.

Can you take the square root of a negative number?

Not in real numbers. The square root of a negative number involves imaginary numbers (using i, where i = sqrt(-1)). For example, sqrt(-4) = 2i. This calculator works with real (non-negative) numbers only.

What does simplified radical form mean?

Simplified radical form expresses a square root with the smallest possible number under the radical sign. For example, sqrt(12) simplifies to 2*sqrt(3) because 12 = 4 * 3, and sqrt(4) = 2. The goal is to extract all perfect square factors from under the radical.

Is sqrt(2) a rational or irrational number?

The square root of 2 is irrational. It cannot be expressed as a fraction of two integers, and its decimal expansion (1.41421356...) never terminates or repeats. This was proven by ancient Greek mathematicians.

How is the square root related to exponents?

The square root of n is equivalent to raising n to the power of 1/2. In mathematical notation, sqrt(n) = n^(1/2). Similarly, a cube root is n^(1/3), and an nth root is n^(1/n).