Cube Root & Square Root Calculator

Lets first understand, what are cube roots & square roots and how we can calculate the cube root & square root of a real number?

Cube Root

A cube root of a number x is a number y such that y³ = x.

All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots and all nonzero complex numbers have three distinct complex cube roots.

The above calculator only calculates the cube root and square root of real numbers.

For example, the real cube root of 64, denoted \(\sqrt[\leftroot{-2}\uproot{2}3]{64}\), is 4, because 4³ = 64, while the other cube roots of 64 are \(-2+2\sqrt{3}i\) and \(-2-2\sqrt{3}i\).

The cube root operation is not distributive with addition or subtraction.

When the number whose cube root is to be taken is a real number, one of the cube roots (the real one) is referred to as the principal cube root, denoted with the radical sign \(\sqrt[\leftroot{-2}\uproot{2}3]{}\).

The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers.

Properties of Cube Root

Real numbers

The cube function is increasing.

The cube function does not give the same result for two different inputs and the cube function covers all real numbers.

The cube function is a bijection, or one-to-one. Then we can define an inverse function that is also one-to-one. For real numbers, we can define a unique cube root of all real numbers, so the cube root of a negative number is a negative number.

A real number has one real cube root and two further cube roots which form a complex conjugate pair.

For instance, the cube roots of 1 are

$$1,\hspace{0.1cm}\frac{-1+\sqrt{3}i}{2},\hspace{0.1cm}\frac{-1-\sqrt{3}i}{2}$$

The last two of these roots lead to a relationship between all roots of any real or complex number. If a number is one cube root of a particular real or complex number, the other two cube roots can be found by multiplying that cube root by one or the other of the two complex cube roots of 1.

Complex numbers

For complex numbers, the principal cube root is usually defined as the cube root that has the greatest real part, or, equivalently, the cube root whose argument has the least absolute value.

It is related to the principal value of the natural logarithm by the formula

$$x^{\frac{1}{3}}=exp\left(\frac{1}{3}\hspace{0.1cm}ln\hspace{0.1cm}x\right)$$

We write x as

$$x=r\hspace{0.1cm}exp(i\theta)$$

where r is a non-negative real number and θ lies in the range \(-\pi < \theta < \pi \),

then the principal complex cube root is

$$\sqrt[\leftroot{-2}\uproot{2}3]{x}=\sqrt[\leftroot{-2}\uproot{2}3]{r}\hspace{0.1cm}exp\left(\frac{i\theta}{3}\right)$$

The cube root of the radius and dividing the polar angle by three in order to define a cube root.

The principal cube root of a negative number is a complex number, for instance \(\sqrt[\leftroot{-2}\uproot{2}3]{-8}\) will not be −2, but rather \(1+\sqrt{3}i\).

Square Root

A cube root of a number x is a number y such that y² = x.

For example, 8 and −8 are square roots of 64, because 8² = (-8)² = 64.

Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by \(\sqrt{x}\), where the symbol \(\sqrt{}\) is called the radix.

For example, the principal square root of 64 is 8, which is denoted by \(\sqrt{64}=8\) because 8² = 8.8 = 64.

The number whose square root is being considered is known as the radicand.

Every positive number x has two square roots: \(\sqrt{x}\) which is positive and \(-\sqrt{x}\) which is negative.

Properties of Square Root

The principal square root function \(f(x)=\sqrt{x}\) is a function that maps the set of nonnegative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length.

The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares.

The square root function is continuous for all nonnegative x and differentiable for all positive x. If f denotes the square root function, whose derivative is given by

$$f'(x)=\frac{1}{2\sqrt{x}}.$$

The square of any positive or negative number is positive and the square of 0 is 0. Therefore, no negative number can have a real square root.

However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number.

This is done by introducing a new number, denoted by i and called the imaginary unit, which is defined such that i² = −1. Using this notation, we can think of i as the square root of −1, but we also have (−i)² = i² = −1 and so \(−i\) is also a square root of −1.

By convention, the principal square root of −1 is i, or if x is any nonnegative number, then the principal square root of \(−x\) is

$$\sqrt{-x}=i\sqrt{x}$$

For calculating cube root & square root using above cube root & square root calculator, you have to just write the number in the given input box and press the calculate button, you will get the result.