Root Formula

The nth root of a number is written using a radical symbol:

This expression asks for a number that, when raised to the power \(n\), gives \(x\). The equivalent exponent form is:

If \(y=\sqrt[n]{x}\), then:

For example, \(\sqrt[3]{64}=4\) because \(4^3=64\). Similarly, \(\sqrt{49}=7\) because \(7^2=49\).

How to Use the Roots Calculator

1. Enter the number under the radical. This is called the radicand and is represented by \(x\).
2. Enter the root index \(n\). Use \(2\) for square root and \(3\) for cube root.
3. Choose the calculation type: nth root, square root, cube root, power check, or exponent form.
4. Choose the number of decimal places you want in the result.
5. Click Calculate Root to see the answer and formula steps.
6. Check the result by raising the root back to the index: if \(y=\sqrt[n]{x}\), then \(y^n=x\).

Root Formulas and Rules

Worked Examples

Example 1: Square root

Find \(\sqrt{81}\).

The principal square root of \(81\) is \(9\). Although both \(9^2=81\) and \((-9)^2=81\), the symbol \(\sqrt{81}\) refers to the principal nonnegative root.

Example 2: Cube root

Find \(\sqrt[3]{64}\).

The cube root of \(64\) is \(4\).

Example 3: Odd root of a negative number

Find \(\sqrt[3]{-125}\).

Odd roots of negative numbers are real. Since \((-5)^3=-125\), the cube root is \(-5\).

Example 4: Fourth root

Find \(\sqrt[4]{16}\).

The principal fourth root of \(16\) is \(2\).

Example 5: Simplify a square root

Simplify \(\sqrt{72}\).

The simplified radical form is \(6\sqrt{2}\). The decimal approximation is about \(8.4853\).

Example 6: Fractional exponent form

Rewrite \(\sqrt[5]{x^2}\) using a fractional exponent.

The denominator of the fractional exponent is the root index, and the numerator is the power.

Complete Guide to Roots

A root is the inverse idea of a power. If powers ask what happens when a number is multiplied by itself repeatedly, roots ask what original number could have produced a given result. For example, \(5^2=25\), so \(\sqrt{25}=5\). Also, \(2^3=8\), so \(\sqrt[3]{8}=2\). Roots and powers are opposite operations in the same way that multiplication and division are related.

The number under the radical symbol is called the radicand. In \(\sqrt[n]{x}\), the radicand is \(x\). The small number written above the radical is called the index. The index tells which root is being taken. If the index is \(2\), the root is a square root. If the index is \(3\), the root is a cube root. If the index is \(4\), the root is a fourth root. The index is not usually written for square roots, so \(\sqrt{x}\) means \(\sqrt[2]{x}\).

Square roots

A square root of a number is a value that gives the number when squared. For example, \(7^2=49\), so \(7\) is a square root of \(49\). Also, \((-7)^2=49\), so \(-7\) is also a square root of \(49\). However, the radical symbol \(\sqrt{49}\) normally means the principal square root, which is the nonnegative root. Therefore, \(\sqrt{49}=7\), not \(\pm7\).

This distinction is important. If you solve the equation \(x^2=49\), then \(x=\pm7\). But if you evaluate \(\sqrt{49}\), the answer is \(7\). Equations and radical expressions have different conventions.

Cube roots

A cube root of a number is a value that gives the number when cubed. For example, \(3^3=27\), so \(\sqrt[3]{27}=3\). Cube roots behave differently from square roots because negative numbers can have real cube roots. Since \((-3)^3=-27\), we have \(\sqrt[3]{-27}=-3\).

This happens because odd powers preserve the sign of the base. A negative number raised to an odd power stays negative. A negative number raised to an even power becomes positive. That is why odd roots of negative numbers are real, while even roots of negative numbers are not real in the real-number system.

Nth roots

The nth root generalizes square roots and cube roots. The expression \(\sqrt[n]{x}\) asks for a number that gives \(x\) when raised to the power \(n\). If \(y=\sqrt[n]{x}\), then \(y^n=x\). This relationship is the central idea behind all root calculations.

For even values of \(n\), a negative radicand has no real nth root. For odd values of \(n\), a negative radicand has a real negative nth root. For example, \(\sqrt[4]{-16}\) is not real, but \(\sqrt[5]{-32}=-2\).

Radicals and fractional exponents

Roots can be written as fractional exponents. This is one of the most important connections in algebra:

This means a square root is the same as raising to the power \(\frac{1}{2}\), a cube root is the same as raising to the power \(\frac{1}{3}\), and a fourth root is the same as raising to the power \(\frac{1}{4}\). More generally:

This rule allows radicals and exponents to be converted back and forth. It is useful in algebra, calculus, exponential functions, logarithms, and scientific modeling.

Principal roots

The principal root is the standard root value returned by the radical symbol. For even roots of positive numbers, the principal root is nonnegative. For example, \(\sqrt{36}=6\), not \(-6\), even though both \(6\) and \(-6\) square to \(36\). When solving equations, you may need both positive and negative solutions. When evaluating the radical expression, you usually use the principal root.

For odd roots, the principal real root follows the sign of the radicand. For example, \(\sqrt[3]{8}=2\) and \(\sqrt[3]{-8}=-2\).

Perfect powers

A perfect power is a number that can be written as another integer raised to a power. Perfect squares include \(1,4,9,16,25,36,49,64,81,100\). Perfect cubes include \(1,8,27,64,125,216,343,512,729,1000\). If a radicand is a perfect power, the root is an integer. For example, \(\sqrt{144}=12\) and \(\sqrt[3]{125}=5\).

Recognizing perfect powers makes root problems faster. Many school questions are designed around common perfect squares and cubes. If the radicand is not a perfect power, the answer may be irrational or may need to be simplified in radical form.

Simplifying radicals

Simplifying a radical means rewriting it so that no perfect power remains inside the radical. For square roots, you look for perfect square factors. For example:

The number \(36\) is a perfect square, so it can come out of the square root as \(6\). The remaining factor \(2\) stays inside the radical.

For cube roots, you look for perfect cube factors. For example:

Simplified radical form is often preferred in exact algebra, while decimal form is useful for measurement, estimation, and applications.

Roots and equations

Roots often appear when solving equations involving powers. If \(x^2=25\), then \(x=\pm5\). If \(x^3=64\), then \(x=4\). If \(x^4=81\), then the real solutions are \(x=\pm3\). When solving equations, remember that even powers often produce two real solutions when the right side is positive.

This is different from evaluating a root expression. The expression \(\sqrt{25}\) equals \(5\), but the equation \(x^2=25\) has solutions \(x=5\) and \(x=-5\).

Roots of negative numbers

Negative numbers require careful attention. If the root index is odd, a negative radicand has a real negative root. For example:

If the root index is even, a negative radicand has no real root. For example, \(\sqrt{-16}\) is not real because no real number squared equals \(-16\). In the complex number system, \(\sqrt{-16}=4i\), where \(i^2=-1\).

Roots and graphs

Root functions have recognizable graph shapes. The square root function \(y=\sqrt{x}\) starts at \((0,0)\) and increases slowly as \(x\) increases. Its domain in real numbers is \(x\ge0\). The cube root function \(y=\sqrt[3]{x}\) is defined for all real numbers and passes through the origin. It can take negative inputs and produce negative outputs.

These graph differences come directly from the rules for even and odd roots. Even root functions are restricted for real inputs when the radicand would become negative. Odd root functions can handle both positive and negative radicands.

Roots in geometry

Square roots appear naturally in geometry because areas use squared units. If a square has area \(A\), its side length is \(\sqrt{A}\). If the area is \(64\) square units, the side length is \(8\) units. The Pythagorean theorem also uses square roots. If \(a^2+b^2=c^2\), then \(c=\sqrt{a^2+b^2}\).

Cube roots appear in volume problems. If a cube has volume \(V\), its side length is \(\sqrt[3]{V}\). If the volume is \(125\) cubic units, the side length is \(5\) units.

Roots in science

Roots are used in physics, chemistry, engineering, statistics, and computer science. In physics, square roots appear in formulas involving energy, velocity, distance, and waves. In statistics, the standard deviation involves a square root because variance is measured in squared units. In engineering, roots can appear when solving equations for dimensions, rates, or magnitudes. In computer science, square roots appear in geometry, graphics, algorithms, and numerical methods.

Exact form vs decimal form

Some roots are exact integers, such as \(\sqrt{100}=10\). Some roots are irrational, such as \(\sqrt{2}\). An irrational root cannot be written as a terminating or repeating decimal. In exact algebra, it is often better to leave the answer as \(\sqrt{2}\). In applied problems, a decimal approximation such as \(1.4142\) may be more useful.

This calculator gives a decimal result and formula steps. For many common square roots, you can also simplify by factoring out perfect squares. In exact math assignments, follow your teacher’s preference for radical form, decimal form, or both.

Common mistakes with roots

One common mistake is writing \(\sqrt{a+b}=\sqrt{a}+\sqrt{b}\). This is not true in general. For example, \(\sqrt{9+16}=\sqrt{25}=5\), but \(\sqrt{9}+\sqrt{16}=3+4=7\). Roots do not distribute over addition.

Another common mistake is forgetting the principal square root convention. The expression \(\sqrt{36}\) equals \(6\), not \(\pm6\). The equation \(x^2=36\) has two solutions, \(x=\pm6\). The difference between evaluating a radical and solving an equation is important.

A third mistake is trying to take an even root of a negative number as if it were real. In real-number algebra, \(\sqrt{-9}\) is not real. In complex-number algebra, it is \(3i\). Always know whether your class or problem is working in the real-number system or complex-number system.

How to check a root answer

The best way to check a root is to raise it back to the index. If you think \(\sqrt[4]{81}=3\), check by calculating \(3^4=81\). If you think \(\sqrt[3]{-216}=-6\), check \((-6)^3=-216\). This check works because roots and powers are inverse operations.

Summary

A root tells which number, raised to a given power, produces the radicand. The nth root is written \(\sqrt[n]{x}\) and can also be written as \(x^{1/n}\). Square roots have index \(2\), cube roots have index \(3\), and higher roots use larger indices. Even roots of negative numbers are not real, while odd roots of negative numbers are real. Roots are used throughout algebra, geometry, science, statistics, and real-world applications. Understanding roots means understanding both radical notation and exponent notation.

Common Mistakes with Roots

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Roots Calculator FAQs