Root Calculator

The expression $$\sqrt[n]{x}$$ is read as the "n^th root of x". It is a number r that, when raised to the power n, yields x. For example, the 4th root of 625 is 5, because 5 raised to the power 4 is 625. Here are some examples to give you an idea:

• The square root of 16 is $$\sqrt{16} = 4$$, since $$4^2 = 16$$.
• The cube root of 27 is $$\sqrt[3]{27} = 3$$, since $$3^3 = 27$$.
• The 5th root of 32 is $$\sqrt[5]{32} = 2$$, since $$2^5 = 32$$.

What are Roots?

Roots are essentially the inverse operation of exponentiation. While exponentiation raises a number to a power, roots work backward to find the base that results in the given value when multiplied by itself a certain number of times.

For example, to find the 3rd root (cube root) of 64, you're asking, “What number multiplied by itself 3 times equals 64?” The answer is $$4$$, because $$4 \times 4 \times 4 = 64$$.

Types of Roots

• Square Root: Denoted by $$\sqrt{x}$$, this is the 2nd root of a number. It answers the question, “What number squared equals x?”
• Cube Root: Represented as $$\sqrt[3]{x}$$, this is the 3rd root of a number, answering, “What number cubed equals x?”
• Higher-Order Roots: For any integer $$n$$, the $$n$$th root is written as $$\sqrt[n]{x}$$, finding the number that satisfies $$r^n = x$$.

Properties of Roots

• Roots can be either positive or negative. For example, $$\sqrt{16}$$ has two solutions: $$4$$ and $$-4$$. However, by convention, the square root symbol ($$\sqrt{}$$) refers to the positive root.
• The root of a fraction, such as $$\sqrt{\frac{1}{9}}$$, is the root of the numerator divided by the root of the denominator: $$\frac{\sqrt{1}}{\sqrt{9}} = \frac{1}{3}$$.
• Roots of negative numbers are undefined in the set of real numbers. For instance, $$\sqrt{-4}$$ is not a real number but is defined in the set of complex numbers as $$2i$$ (where $$i$$ is the imaginary unit).

Why Are Roots Useful?

Roots have applications in various fields, including geometry, physics, and data science. For example:

• In geometry, calculating the diagonal of a square involves the square root function.
• In physics, the root is used in formulas like the Pythagorean theorem or calculating RMS (root mean square) values.
• In statistics, roots are often part of standard deviation calculations and variance analyses.

Using our root calculator, you can easily compute roots for any number and any degree. Whether it's a square root, cube root, or a higher-order root, the tool simplifies the process for you.