Free Radical Calculator

Radical Calculator (also referred to as an nth root calculator or radical expression calculator) is a free online tool that quickly computes any root of a real number. Its interface is built around three fields: the number (radicand), the root degree, and the resulting root. You supply any two, and the calculator instantly determines the third. This makes it just as convenient for finding a square root, a cube root, or any higher-order root, and equally useful for reverse calculations (e.g., finding the radicand when you know the result and the degree). For anyone who needs to verify simplifications, the tool also acts as a simplify radicals calculator.

Radical Notation and the Definition of an nth Root

The symbol $\sqrt[n]{x}$ represents the $n$-th root of $x$. By definition, if $r = \sqrt[n]{x}$, then

$$r^{n} = x,$$

which is equivalent to $r = x^{1/n}$. Therefore, extracting a radical is the inverse operation of exponentiation. For instance, $\sqrt[3]{8} = 2$ because $2^{3}=8$, and $\sqrt[4]{81}=3$ because $3^{4}=81$. This relationship between radicals and fractional exponents is the key to simplifying many algebraic operations.

Conditions for Using Radicals

To obtain a real result, the radicand should generally be non-negative when the root degree is even. For odd degrees, it is possible to take the root of a negative number and still get a real answer (e.g., $\sqrt[3]{-27} = -3$). However, allowing negative inputs can lead to multiple complex roots, so this root calculator restricts the radicand to non-negative numbers for all degrees. The degree $n$ is typically a positive integer, but the tool also accepts fractional and negative exponents because a radical can always be expressed as a power.

How to Use This nth Root Calculator

The three fields are clearly labeled:

• Number ($x$) – the value under the radical sign.
• Radical degree ($n$) – the index of the root.
• Radical (the result $r$) – the $n$-th root of $x$.

Fill in any two fields and the missing one is computed automatically. For example, to find $\sqrt[5]{32}$, enter $32$ and $5$; the result $2$ appears instantly. You can also enter the result and the degree to retrieve the original number, or enter the number and the result to obtain the required degree. This flexibility makes the calculator useful for checking homework, solving equations, and exploring the inverse relationship between exponents and radicals.

Adding and Subtracting Radicals

Radicals can be added or subtracted only when they are like radicals – that is, they share the same degree and the same radicand. For example:

$$\sqrt[4]{7} + 2\sqrt[4]{7} = 3\sqrt[4]{7}, \qquad 5\sqrt[3]{5} - 3\sqrt[3]{5} = 2\sqrt[3]{5}$$

If the radicals do not appear to be like, check whether one of them can be simplified. Consider $\sqrt[3]{192} + \sqrt[3]{3}$. Because $192 = 64 \times 3$, we have:

$$\sqrt[3]{192} = \sqrt[3]{64 \times 3} = \sqrt[3]{64} \times \sqrt[3]{3} = 4\sqrt[3]{3}$$

The sum then becomes $4\sqrt[3]{3} + \sqrt[3]{3} = 5\sqrt[3]{3}$. The same procedure works for subtraction. Another example: $\sqrt{72} + \sqrt{2}$. Since $72 = 36 \times 2$,

$$\sqrt{72} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2},$$

so $\sqrt{72} + \sqrt{2} = 6\sqrt{2} + \sqrt{2} = 7\sqrt{2}$. Always check for simplification before adding or subtracting radicals.

Multiplying Radicals

Multiplying radicals is simpler than addition. When two radicals have the same degree, multiply the radicands and keep the same root index:

$$\sqrt[4]{7} \times \sqrt[4]{4} = \sqrt[4]{7 \times 4} = \sqrt[4]{28}$$

If coefficients appear in front of the radicals, multiply them separately:

$$4\sqrt[3]{3} \times 2\sqrt[3]{5} = (4 \times 2) \sqrt[3]{3 \times 5} = 8\sqrt[3]{15}$$

A more involved example: $3\sqrt[4]{5} \times 2\sqrt[4]{20} = 6\sqrt[4]{100}$. Since $100 = 10^{2}$, we can simplify $\sqrt[4]{100} = 10^{2/4} = 10^{1/2} = \sqrt{10}$, so the result is $6\sqrt{10}$.

When the radicals have different indices but the same radicand, convert each to a fractional exponent:

$$\sqrt[4]{2} \times \sqrt[3]{2} = 2^{1/4} \times 2^{1/3} = 2^{1/4 + 1/3} = 2^{7/12} = \sqrt[12]{2^{7}} = \sqrt[12]{128}$$

Thus, any multiplication of radicals can be handled by switching between radical notation and exponent notation.

Dividing Radicals

Division follows the same logic. With the same degree, divide the radicands:

$$\sqrt[5]{8} \div \sqrt[5]{4} = \sqrt[5]{8 \div 4} = \sqrt[5]{2}$$

If coefficients are present, divide them as well:

$$5\sqrt[3]{15} \div 2\sqrt[3]{5} = \frac{5}{2} \sqrt[3]{15 \div 5} = 2.5\sqrt[3]{3}$$

For radicals with different indices but the same radicand, convert to fractional exponents and subtract the exponents:

$$\sqrt[4]{6} \div \sqrt[5]{6} = 6^{1/4} \div 6^{1/5} = 6^{1/4 - 1/5} = 6^{1/20} = \sqrt[20]{6}$$

Always inspect the final radical to see if it can be simplified further.

Simplifying Radicals

Simplifying a radical means factoring the radicand so that any perfect power of the index is taken outside the radical. For example, $\sqrt[3]{24}$ can be written as $\sqrt[3]{8 \times 3} = \sqrt[3]{8} \times \sqrt[3]{3} = 2\sqrt[3]{3}$. Similarly,

$$\sqrt[3]{54} = \sqrt[3]{27 \times 2} = 3\sqrt[3]{2}, \qquad \sqrt[4]{162} = \sqrt[4]{81 \times 2} = 3\sqrt[4]{2}$$

The simplify radicals calculator aspect of the tool can help you verify whether a radical is in its simplest form or can be reduced further.

The Radical Function and Its Graph

The most basic radical function is $f(x) = \sqrt[n]{x}$, where $n$ is a positive integer. For $n=2$ this is the square root function, whose graph is a half‑parabola turned 90 degrees; as $n$ increases, the curve becomes flatter near the origin.

A more general radical function includes scaling and translation parameters:

$$f(x) = a \times \sqrt[n]{b \times x - h} + k$$

• $a$ stretches or compresses the graph vertically.
• $b$ stretches or compresses it horizontally.
• $h$ shifts the graph to the left or right.
• $k$ shifts it up or down.

These parameters allow you to model a wide variety of radical growth and decay scenarios. While the calculator itself focuses on numeric computation, understanding the effect of each parameter is essential for graphing and transformations.