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Root Calculator

How to use Root Calculator
  1. Pick a mode with the three tabs at the top: Calculate finds one root, Laws applies a radical law to your numbers, Compare sets two radicals side by side. Learn more about the modes
  2. In Calculate mode, click Square for a square root. Learn more about square roots
  3. Click Cube for a cube root; negative numbers are allowed here. Learn more about cube roots
  4. Click Custom and type the index in the small box for any other root. Learn more about nth roots
  5. Type the number under the radical sign, or click a chip under the buttons to load a perfect square, cube or power. Learn more about the number
  6. The result appears after the equals sign as you type; the Result card lists it in several forms. Click Calculate to keep it in the Recent strip. Learn more about the results
  7. The How this works panel on the right explains the current root, law or comparison with your own numbers; its link opens the section below that explains the topic in full. Learn more about the panel
  8. In Laws mode, click a law on the left, edit its boxes, and read the numbered steps. Learn more about Laws mode
  9. In Compare mode, type an index and a number for expression A and for expression B, then read the verdict. Learn more about Compare mode
  10. Click Reset to return to the square root of 25. The result reads imaginary for an even root of a negative number. Learn more about getting started
Compute a root
Root type
√=5
Result
Square root of 25
Exact
5
Decimal
5
Scientific
5.0000e+0
Fractional exponent
Neighboring roots (click to set)
4.69
4.80
4.90
5
5.10
5.20
5.29
Apply a radical law
√·√
Step-by-step
1
2
3
Compare two radicals
Expression A
√
= 7.0711
vs
Expression B
√
= 5.0000
Verdict
A is larger — about 1.414× the other.
Decimal A
7.071068
Decimal B
5.000000
Ratio A/B
1.4142
A − B
2.0711
Common index
6
A restated
B restated
How this works

Square Root

The number that multiplied by itself gives the radicand: 25=5\sqrt{25} = 5. A negative radicand has no real square root. Learn more about square roots · All root types

What just happened

25 is a perfect square: .

The concept

The square root of x asks: what number, multiplied by itself, gives x? Formally, means with .

Square roots near 25
nExpressionValue
204.472136
214.582576
224.690416
234.795832
244.898979
255
265.09902
275.196152
285.291503
295.385165
305.477226
315.567764































Getting Started with the Root Calculator

DemoThe three modes and Reset
Step 0 of 5
The calculator has three modes, picked with the tabs at the top: Calculate, Laws and Compare. Calculate finds one root and shows it in several forms. Laws applies one radical law to numbers you choose and lists the steps. Compare sets two radicals side by side and says which is larger. Each mode keeps its own numbers, so you can switch tabs without losing anything. The demo above clicks through them and ends with Reset.

In Calculate mode, start with the Root type row: Square and Cube fix the index at 22 or 33, and Custom opens a small box for any other index. A row of chips under the buttons offers perfect squares, perfect cubes or powers of 22, so a click gives a whole-number result.

Type the number under the radical sign in the large box. Whole numbers, decimals and, for odd indices, negative numbers all work. The result appears at once after the equals sign; the calculator starts with 25=5\sqrt{25} = 5, and 3636 gives 36=6\sqrt{36} = 6.

Click Calculate to keep the current root in the Recent strip, and Reset to return to 25\sqrt{25}. What a root is, and why even roots of negative numbers are not real, is explained in what are roots.

Using Square Root Mode

DemoSquare roots: perfect, irrational and impossible
Step 0 of 5
Click Square to take square roots. The radical sign shows no index, and the chips under the buttons offer the perfect squares from 44 to 169169. A square root asks: what number multiplied by itself equals the radicand? For 25\sqrt{25} the answer is 55, because 5×5=255 \times 5 = 25.

Type 3636 and the result reads 66 at once; the Result card adds the Exact form because 3636 is a perfect square. Type 5050 and the result is 50≈7.0711\sqrt{50} \approx 7.0711, with the Simplified radical form 525\sqrt{2} below it. Click the 144144 chip for 144=12\sqrt{144} = 12, and type 22 for the famous 2≈1.4142\sqrt{2} \approx 1.4142, which the panel marks as irrational.

A negative number has no real square root. Type −25-25 and the result reads imaginary; the Result card's status says Imaginary (no real root) and the panel explains why. Decimals work: 0.25=0.5\sqrt{0.25} = 0.5.

The reasoning behind the principal root and the rule for negatives is in square roots.

Using Cube Root Mode

DemoCube roots, including negative numbers
Step 0 of 5
Click Cube to take cube roots. The radical sign shows the index 33, and the chips offer the perfect cubes from 88 to 10001000. A cube root asks: what number multiplied by itself twice equals the radicand? For 273\sqrt[3]{27} the answer is 33, because 3×3×3=273 \times 3 \times 3 = 27. With the starting number 2525, the cube root is 253≈2.9240\sqrt[3]{25} \approx 2.9240.

Type 2727 for 33, click the 10001000 chip for 1010, and type 1010 for 103≈2.1544\sqrt[3]{10} \approx 2.1544, a decimal approximation because 1010 is not a perfect cube.

Unlike square roots, cube roots accept negative numbers: type −8-8 and the result is −2-2, because (−2)×(−2)×(−2)=−8(-2) \times (-2) \times (-2) = -8; the panel notes that a negative cube root is fine. The same holds for every odd index.

Why odd roots of negative numbers exist while even ones do not is explained in cube roots.

Using Nth Root Mode

DemoAny index with Custom
Step 0 of 5
Click Custom to take a root of any index. A small box appears to the left of the radical sign holding the index; it starts at 22, so the result is unchanged until you type another index. With the starting number 2525 and index 44, the result is 254≈2.2361\sqrt[4]{25} \approx 2.2361. The chips under the buttons change to the powers of 22 from 22 to 256256.

Type 44 as the index and 8181 under the radical to get 33, because 34=813^4 = 81. Type 55 and 3232 to get 22, because 25=322^5 = 32. The index must be a whole number greater than 00; a 11 returns the radicand itself.

For an even index a negative radicand gives imaginary, as with square roots. For an odd index it is allowed: index 55 with −32-32 gives −2-2, because (−2)5=−32(-2)^5 = -32.

The index can be large: 102410=2\sqrt[10]{1024} = 2. How the index decides the sign rules is explained in nth roots.

Reading Results and Understanding Output

DemoResult forms, Recent, neighbors and messages
Step 0 of 5
The result appears after the equals sign as soon as the radicand and the index are valid; you do not have to click anything. Whole-number results print in full, 49=7\sqrt{49} = 7; others show four decimal places, 50≈7.0711\sqrt{50} \approx 7.0711.

The Result card below the form lists the value in several forms: Exact when the radicand is a perfect power, Decimal to six places, Scientific, the Simplified radical when a perfect power can be factored out, 50=52\sqrt{50} = 5\sqrt{2}, and the Fractional exponent form, 501/250^{1/2}. The copy icon next to each form copies it.

Under the forms, Neighboring roots shows the radicands from three below to three above the current one with their roots; click one to load it, for example 4949 next to 5050, which gives the exact value 77. Calculate stores the current root in the Recent strip; click an entry to bring it back. The reference table on the right lists the perfect powers near your number for the current index.

The result line reads enter a value while the box is empty, imaginary for an even root of a negative number, and undefined for a value beyond the computer's range; the Result card's status says Waiting for input, Imaginary (no real root) or the matching reason. The imaginary case is explained in square roots.

Using the Explanation Panel

DemoWhat the panel says for different numbers
Step 0 of 5
The How this works card at the top of the right-hand column changes with every click and keystroke. It opens with a short note on the current state: the root type, the law or the comparison you have picked, each ending with a link to the section on this page that explains it in full.

In Calculate mode the card then writes out the root for your numbers as a power equation, points out what is special about them, and states the concept behind the chosen root type. The special cases it flags are the root of zero, the root of one, an irrational result such as 10≈3.1623\sqrt{10} \approx 3.1623, a negative radicand under a cube root, and a radical that can be simplified, such as 50=52\sqrt{50} = 5\sqrt{2}. Switching to Cube changes the concept text and the computation: 503≈3.6840\sqrt[3]{50} \approx 3.6840.

In Laws mode the card names the law, explains why it holds and shows a common mistake. In Compare mode it explains how the two radicals are compared.

Below the card, Calculate mode also shows a reference table of perfect powers near your number; click a row to load that radicand. The laws the panel refers to are collected in properties of roots.

Using Laws Mode

DemoProduct, quotient, power and nested radicals
Step 0 of 5
Click the Laws tab to apply one radical law to numbers of your own. The left card lists six laws as buttons, each with its formula: product rule, quotient rule, power of a root, nested radicals, rationalize and exponent form. The product rule is selected when you open the mode.

Under the buttons, the input row shows the boxes the law needs: two radicands aa and bb and an index nn for the product and quotient rules, a radicand, a power mm and an index for the power of a root, two indices for nested radicals, and one radicand and index for rationalizing and the exponent form. The boxes start with sample values, so a result is visible at once: 12⋅3=36=6\sqrt{12} \cdot \sqrt{3} = \sqrt{36} = 6, and with a=27a = 27 the product becomes 81=9\sqrt{81} = 9. The quotient rule starts with 50/2=25=5\sqrt{50} / \sqrt{2} = \sqrt{25} = 5, the power of a root with (4)3=64=8(\sqrt{4})^3 = \sqrt{64} = 8, and nested radicals with 643=646=2\sqrt{\sqrt[3]{64}} = \sqrt[6]{64} = 2.

The Step-by-step list under the inputs rewrites your numbers with the law: the starting expression, the law applied, the value, and a direct check. Change a box and the steps follow at once.

The How this works card on the right explains the selected law and warns about its most common mistake. All six laws are summarised in properties of roots.

Using Compare Mode

DemoCompare two radicals
Step 0 of 4
Click the Compare tab to set two radicals side by side. The card shows Expression A and Expression B, each with a small index box and a radicand box, and prints the value of each as you type. The mode opens with 50≈7.0711\sqrt{50} \approx 7.0711 against 1253=5\sqrt[3]{125} = 5, so A is larger, about 1.4141.414 times.

The Verdict line says which expression is larger, or that they are equal, and gives the ratio. Under it, a row of figures lists both values to six decimals, the ratio A/BA / B and the difference A−BA - B; when the indices differ, it also restates both radicals over a common index.

Type 4949 under the radical of A for 49=7\sqrt{49} = 7 against 55; type 343343 under the radical of B and 3433=7\sqrt[3]{343} = 7 makes the two equal; change the index of B to 22 and 343≈18.5203\sqrt{343} \approx 18.5203 makes B the larger one.

An even root of a negative number shows a dash, since it has no real value. How radicals with different indices are compared is explained in comparing two radicals.

What Are Roots

A root is a value that, when multiplied by itself a certain number of times, produces a given number. Roots represent the inverse operation of exponentiation. If 23=82^3 = 8, then the cube root of 88 is 22. Roots "undo" powers.

Mathematicians express roots using radical notation with the radical symbol √. The radicand is the number under the symbol whose root you're finding. The index or degree is the small number indicating how many times to multiply. In 273\sqrt[3]{27}, the radicand is 2727 and the index is 33.

When no index appears, it's implicitly 22, meaning square root. Writing √1616 is the same as 162\sqrt[2]{16}. Both mean "what squared equals 1616?" The answer is 44. For cube roots, you write the 33 explicitly: 83=2\sqrt[3]{8} = 2.

Roots connect to exponents through fractional powers. The square root of xx equals x1/2x^{1/2}. The cube root of xx equals x1/3x^{1/3}. The nth root of xx equals x1/nx^{1/n}. This connection lets calculators and computers compute roots using exponentiation: 814=811/4=3\sqrt[4]{81} = 81^{1/4} = 3.

In Calculate mode the three buttons pick the index: square roots, cube roots, or any index with nth roots. The definition is in what is a root, the notation in radical and exponent notation, and the link to powers in connection to powers.

Square Roots

The square root of a non-negative number aa is the non-negative number whose square is aa: 25=5\sqrt{25} = 5 because 52=255^2 = 25. Perfect squares give whole numbers, 144=12\sqrt{144} = 12; every other positive number gives an irrational decimal, 2≈1.4142\sqrt{2} \approx 1.4142 and 50≈7.0711\sqrt{50} \approx 7.0711.

Two numbers square to 2525, namely 55 and −5-5, but the radical sign means the principal root, the non-negative one. The solutions of x2=25x^2 = 25 are written x=±25x = \pm\sqrt{25}. A square root is often left exact and simplified instead of rounded: 50=25×2=52\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}.

No real number squares to a negative number, since a square is never negative, so −25\sqrt{-25} is not real; the calculator reports it as imaginary. In the complex numbers it is 5i5i. The common mistake is to write 25=±5\sqrt{25} = \pm 5: the radical itself is 55, and the ±\pm belongs to the equation.

See square roots, cube roots, and beyond, principal root convention and even index radicals.

Cube Roots

The cube root of aa is the number whose cube is aa: 273=3\sqrt[3]{27} = 3 because 33=273^3 = 27, and 10003=10\sqrt[3]{1000} = 10. For a number that is not a perfect cube the root is an irrational decimal: 103≈2.1544\sqrt[3]{10} \approx 2.1544 and 253≈2.9240\sqrt[3]{25} \approx 2.9240.

Every real number has exactly one real cube root, negative numbers included, because a cube keeps the sign of its base: −83=−2\sqrt[3]{-8} = -2 since (−2)3=−8(-2)^3 = -8. There is no principal-root choice to make, as there is for even indices, and no imaginary case.

Cube roots turn a volume back into an edge length: a cube of volume 2727 cubic units has edge 33. The common mistake is to treat −83\sqrt[3]{-8} like −8\sqrt{-8} and declare it impossible, or to halve instead of taking a root: 10003\sqrt[3]{1000} is 1010, not 500500.

See odd index radicals, when radicands are negative and the cube root function.

Nth Roots

The nn-th root of aa is the number whose nn-th power is aa, written an\sqrt[n]{a} with the index nn in the crook of the radical sign: 814=3\sqrt[4]{81} = 3 because 34=813^4 = 81, 325=2\sqrt[5]{32} = 2 because 25=322^5 = 32, 102410=2\sqrt[10]{1024} = 2. An index of 22 is a square root and is usually left unwritten; an index of 11 returns the number itself.

The sign rules follow the parity of the index. For an even nn the radicand must be non-negative and the principal root is the non-negative one; −164\sqrt[4]{-16} is not real. For an odd nn every real radicand works and the root keeps its sign: −325=−2\sqrt[5]{-32} = -2.

By the exponent form, an=a1/n\sqrt[n]{a} = a^{1/n}, which is how the calculator evaluates any index: 254=250.25≈2.2361\sqrt[4]{25} = 25^{0.25} \approx 2.2361. The common mistake is to divide the radicand by the index: 814\sqrt[4]{81} is 33, not 20.2520.25.

See square roots, cube roots, and beyond, domain and range by index and general nth root functions.

Properties of Roots

Product property: The root of a product equals the product of roots: ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b}. For example, 36=4×9=4×9=2×3=6\sqrt{36} = \sqrt{4 \times 9} = \sqrt{4} \times \sqrt{9} = 2 \times 3 = 6. This works for all roots: 8×273=83×273=2×3=6\sqrt[3]{8 \times 27} = \sqrt[3]{8} \times \sqrt[3]{27} = 2 \times 3 = 6.

Quotient property: The root of a quotient equals the quotient of roots: ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}. For example, 254=254=52=2.5\sqrt{\frac{25}{4}} = \frac{\sqrt{25}}{\sqrt{4}} = \frac{5}{2} = 2.5. This property helps simplify complex radical expressions.

Power property: Taking a root and raising to a power are inverse operations: (an)n=a(\sqrt[n]{a})^n = a and ann=a\sqrt[n]{a^n} = a (for non-negative aa and even nn). For example, (5)2=5(\sqrt{5})^2 = 5 and 32=3\sqrt{3^2} = 3. This relationship shows how roots "undo" exponents.

Simplification: Roots of perfect powers simplify to whole numbers. 49=7\sqrt{49} = 7, 1253=5\sqrt[3]{125} = 5, 164=2\sqrt[4]{16} = 2. Non-perfect powers remain as decimals or can be simplified using the product property: 50=25×2=25×2=52\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}.

Laws mode also covers nested radicals, rationalizing the denominator and the exponent form of a root. Every law is an exponent law in disguise, because a root is a fractional power. The rules are collected in rules for radicals and summary of radical rules; simplification is taught in factoring out perfect powers and operations strategy.

The Product Rule for Radicals

Roots with the same index multiply under one radical: an⋅bn=abn\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}, for radicands that are allowed under that index.

Example: 12⋅3=36=6\sqrt{12} \cdot \sqrt{3} = \sqrt{36} = 6, and with 2727 instead of 1212, 27⋅3=81=9\sqrt{27} \cdot \sqrt{3} = \sqrt{81} = 9. The rule is the exponent law a1/nb1/n=(ab)1/na^{1/n} b^{1/n} = (ab)^{1/n} in radical dress.

Read from right to left it simplifies radicals: 50=25⋅2=25⋅2=52\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}, which the calculator shows as the simplified radical form. The common mistake is to multiply radicals of different indices under one sign: 2⋅23\sqrt{2} \cdot \sqrt[3]{2} is not 4\sqrt{4}; the indices must match first. Another is to apply the rule to a sum: a+b\sqrt{a + b} is not a+b\sqrt{a} + \sqrt{b}, since 9+16=5\sqrt{9 + 16} = 5 while 3+4=73 + 4 = 7.

See product rule and multiplying radicals with the same index.

The Quotient Rule for Radicals

Roots with the same index divide under one radical: anbn=abn\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\dfrac{a}{b}}, with b≠0b \neq 0.

Example: 50/2=25=5\sqrt{50} / \sqrt{2} = \sqrt{25} = 5. In the other direction, 254=254=52\sqrt{\dfrac{25}{4}} = \dfrac{\sqrt{25}}{\sqrt{4}} = \dfrac{5}{2}. The rule is the exponent law a1/n/b1/n=(a/b)1/na^{1/n} / b^{1/n} = (a/b)^{1/n}.

It is used to take roots of fractions one part at a time and, together with rationalizing, to clear roots out of denominators. The common mistake is to split a root over a difference: a−b\sqrt{a - b} is not a−b\sqrt{a} - \sqrt{b}; 25−9=4\sqrt{25 - 9} = 4 while 5−3=25 - 3 = 2. As with the product rule, the indices must agree.

See quotient rule and dividing radicals.

Power of a Root

A root raised to a power equals the root of the power: (an)m=amn=am/n(\sqrt[n]{a})^m = \sqrt[n]{a^m} = a^{m/n}.

Example: (4)3=23=8(\sqrt{4})^3 = 2^3 = 8, and 43=64=8\sqrt{4^3} = \sqrt{64} = 8 as well; both equal 43/24^{3/2}. Likewise (83)2=4=82/3(\sqrt[3]{8})^2 = 4 = 8^{2/3}. The special case m=nm = n undoes the root: (an)n=a(\sqrt[n]{a})^n = a, and for an odd index ann=a\sqrt[n]{a^n} = a too, while for an even index a2=∣a∣\sqrt{a^2} = |a|, so (−3)2=3\sqrt{(-3)^2} = 3.

The rule lets you take the root first, when the radicand is a perfect power, and raise the small number instead of the large one. The common mistake is to confuse the two numbers in the fraction: am/na^{m/n} has the power mm on top and the index nn below, so 43/2=84^{3/2} = 8, not 42/34^{2/3}.

See power rule and the identity for even roots.

Nested Radicals

A root of a root is a single root whose index is the product of the two indices: amn=anm\sqrt[n]{\sqrt[m]{a}} = \sqrt[nm]{a}.

Example: 643=646=2\sqrt{\sqrt[3]{64}} = \sqrt[6]{64} = 2, because 26=642^6 = 64; checking the long way, 643=4\sqrt[3]{64} = 4 and 4=2\sqrt{4} = 2. The rule is the exponent law (a1/m)1/n=a1/(mn)(a^{1/m})^{1/n} = a^{1/(mn)}. The order of the two roots does not matter: 643=83=2\sqrt[3]{\sqrt{64}} = \sqrt[3]{8} = 2 as well.

The common mistake is to add the indices: 643\sqrt{\sqrt[3]{64}} is not 645≈2.2974\sqrt[5]{64} \approx 2.2974. For even indices the inner radicand must be non-negative at each stage.

See radical of a radical.

Rationalizing the Denominator

Rationalizing rewrites a fraction so that no root remains in the denominator, by multiplying top and bottom by the same radical: 12=12⋅22=22≈0.7071\dfrac{1}{\sqrt{2}} = \dfrac{1}{\sqrt{2}} \cdot \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{\sqrt{2}}{2} \approx 0.7071.

For a higher index, multiply by the power that completes the radicand: 123=4383=432≈0.7937\dfrac{1}{\sqrt[3]{2}} = \dfrac{\sqrt[3]{4}}{\sqrt[3]{8}} = \dfrac{\sqrt[3]{4}}{2} \approx 0.7937. In general, 1an=an−1na\dfrac{1}{\sqrt[n]{a}} = \dfrac{\sqrt[n]{a^{n-1}}}{a}. The value does not change, since the fraction was multiplied by 11.

Rationalized forms were the standard before calculators, because dividing by a decimal root by hand is awkward, and they are still the expected simplest form. The common mistake is to multiply only the denominator: 12\dfrac{1}{\sqrt{2}} is not 12\dfrac{1}{2}.

See rationalizing monomial denominators and rationalizing higher-index denominators.

Exponent Form

Every root is a fractional power: an=a1/n\sqrt[n]{a} = a^{1/n}. 83=81/3=2\sqrt[3]{8} = 8^{1/3} = 2, 814=810.25=3\sqrt[4]{81} = 81^{0.25} = 3, 50=501/2\sqrt{50} = 50^{1/2}, which the calculator lists as the fractional exponent form.

The exponent form is why every radical law is an exponent law: the product rule is a1/nb1/n=(ab)1/na^{1/n} b^{1/n} = (ab)^{1/n}, the power of a root is (a1/n)m=am/n(a^{1/n})^m = a^{m/n}, and nested radicals are (a1/m)1/n=a1/(mn)(a^{1/m})^{1/n} = a^{1/(mn)}. Combining a power and a root gives a general rational exponent: 823=82/3=4\sqrt[3]{8^2} = 8^{2/3} = 4.

It is also how computers evaluate roots, through logarithms and exponentials. The common mistake is to read a1/na^{1/n} as a/na / n: 81/38^{1/3} is 22, not 8/38/3.

See radical and exponent notation, unit fraction exponents and converting between forms.

Comparing Two Radicals

Two radicals with the same index compare like their radicands: 50>49\sqrt{50} > \sqrt{49} because 50>4950 > 49, since the root function increases.

With different indices, bring both to a common index first. For 50\sqrt{50} against 1253\sqrt[3]{125} the common index is 66: 50=5036=1250006\sqrt{50} = \sqrt[6]{50^3} = \sqrt[6]{125000} and 1253=12526=156256\sqrt[3]{125} = \sqrt[6]{125^2} = \sqrt[6]{15625}, so the square root is larger. The values agree: 7.07117.0711 against 55, a ratio of about 1.4141.414. The exponent form gives the same test with fractions: compare 501/250^{1/2} with 1251/3125^{1/3} by raising both to the 66-th power.

The common mistake is to compare radicands across indices: 3433=7\sqrt[3]{343} = 7 is far smaller than 343≈18.5203\sqrt{343} \approx 18.5203 although the radicands are equal, because a higher index pulls the value down faster.

See multiplying radicals with different indices and combining radicals via exponents.

Types of Roots Explained

Square roots are the most common roots. The square root of xx asks: "What number squared equals xx?" Since 72=497^2 = 49, we know 49=7\sqrt{49} = 7. Square roots only work with non-negative numbers in real mathematics. Every positive number has two square roots (positive and negative), but calculators typically show only the principal (positive) root.

Cube roots involve three multiplications. The cube root of xx asks: "What number cubed equals xx?" Since 43=644^3 = 64, we know 643=4\sqrt[3]{64} = 4. Unlike square roots, cube roots work with negative numbers: −273=−3\sqrt[3]{-27} = -3 because (−3)3=−27(-3)^3 = -27. Every real number has exactly one real cube root.

Fourth roots and higher follow the same pattern. The fourth root asks about four multiplications, the fifth about five, and so on. Even roots (2nd, 4th, 6th) behave like square roots—they don't accept negative inputs. Odd roots (3rd, 5th, 7th) behave like cube roots—they accept negative inputs.

Principal roots are the primary values calculators return. For even roots of positive numbers, this is the positive root. For 16\sqrt{16}, both 44 and −4-4 work mathematically (42=164^2 = 16 and (−4)2=16(-4)^2 = 16), but calculators show 44 as the principal root. The even and odd cases are compared in even vs odd at a glance, and the root functions in parent functions at a glance.