Калькулятор квадратного корня

Square root, cube root and nth root

A root is the inverse operation of a power. The square root √a is the number that, multiplied by itself, gives a. The cube root ³√a is the number that, raised to the third power, gives a. In general, the nth root corresponds to the power a^(1/n). The cube root and other odd roots are also defined for negative numbers, whereas the square root of a negative number is not computed as a real number.

Как пользоваться калькулятором

1. Enter the number under the root. The number whose root is calculated.
2. Set the degree of the root n. 2 = square root, 3 = cube root, or any integer.
3. Use the shortcuts. The square root, cube root and 4th root are available as buttons.
4. Calculate. For the square root, the simplified form is also shown, if there is one.

Why is √50 = 5√2?

A square root can often be simplified by separating the largest square factor from the radicand. Since 50 = 25 × 2 and 25 is the square of 5, we get √50 = √25 × √2 = 5√2. This exact form is useful in mathematics, where the answer is often wanted in radical form rather than as an approximation. The calculator does the simplification automatically and shows both the exact and the approximate value.

Example: simplifying √72

Let's simplify √72. Find the largest square factor: 72 = 36 · 2, and 36 = 6². Thus √72 = √36 · √2 = 6√2 ≈ 8.485. If you don't immediately spot the largest square factor, you can break it up into smaller parts: 72 = 4 · 18 = 4 · 9 · 2, so √4 = 2 and √9 = 3, i.e. 2 · 3 · √2 = 6√2. The end result is the same. The calculator gives both the exact radical form and the decimal approximation.

Roots and powers together

A root is a power with a fractional exponent: √a = a^(1/2), ³√a = a^(1/3) and, in general, the nth root is a^(1/n). This connection makes the calculations consistent: for example, (√a)² = a^(1/2·2) = a¹ = a. The fractional exponent unites the root and the power, so √(a³) can be written as a^(3/2). Once you grasp this, handling radical expressions becomes easier and you see why the rules for roots and powers are the same.

Irrational numbers and approximations

The square root of many numbers is not an exact decimal but an irrational number whose decimal representation continues indefinitely without a repeating pattern. For example, √2 ≈ 1.41421356… never ends and never repeats. For this reason, in mathematics the answer is often left in exact radical form, such as 5√2, rather than being rounded. The radical form is exact, whereas the decimal is always an approximation.

The ancient Greeks already noticed that the diagonal of a square cannot be expressed as commensurable with the side — this is precisely what led to the concept of irrational numbers. In practical measurements an approximation is enough, but in calculation it is worth keeping the exact form as long as possible and rounding only at the end. This calculator shows both the simplified radical form and the decimal approximation, so you get either as needed.

Good to know about roots

• The square root always returns a non-negative value: √9 = 3, not −3.
• The root of a product is the product of the roots: √(a·b) = √a · √b, which makes simplification possible.
• Odd roots (such as the cube root) are also defined for negative numbers.
• A root can always be written as a power with a fractional exponent: ⁿ√a = a^(1/n).

Подводя итог

The square root calculator gives the square root, cube root and nth root as well as the simplified radical form. Enter the number under the root and the degree of the root, and you get both the exact form and the decimal approximation. For square roots, the tool separates the largest square factor, so the answer is presented in the exact form wanted in mathematics.

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