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Root Calculator

?How to use Root Calculator+
  • • Select root type: square, cube, or nth root
  • • For nth root, specify the value of n in the small input box
  • • Enter the number you want to find the root of
  • • Click Calculate to see the result
  • • Use Reset to clear all fields and start over

√
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A square root of a number is a value that, when multiplied by itself, gives the number. For example, the square root of $25$ is $5$, because $5 \times 5 = 25$.



Getting Started with the Root Calculator
Using Square Root Mode
Using Cube Root Mode
Using Nth Root Mode
Reading Results and Understanding Output
Using the Explanation Panel
What Are Roots
Types of Roots Explained
Properties of Roots
Related Calculators and Mathematical Tools




























Getting Started with the Root Calculator

The root calculator has three main areas: a top selection bar with three root type buttons, a central calculation display showing your inputs and result, and a right sidebar with explanations. Start by looking at the three large buttons at the top labeled Square Root, Cube Root, and Nth Root. Click any button to select that root type.

After selecting your root type, examine the calculation display in the blue-bordered box. You'll see the radical symbol (√ or ∛), an input field for your number, an equals sign, and a result area. The display changes based on which root type you selected—Square Root shows √, Cube Root shows ∛, and Nth Root adds an extra small input box for the n value.

Enter your number in the main input field. Type any positive or negative number including decimals like 2525, −8-8, or 15.515.5. A small blue question mark appears in the corner of each input box. Hover over these question marks to see helpful tooltips explaining what each field does.

The Calculate button at the bottom processes your input and displays the result in the rightmost box. The Reset button clears all fields and returns the calculator to its starting state. The right sidebar updates automatically when you change root types, showing explanations and links relevant to your current selection.

Using Square Root Mode

Click the Square Root button at the top to activate square root mode. The display shows the √ symbol and a single input field. Square roots ask: "What number multiplied by itself equals my input?" For 25\sqrt{25}, the answer is 55 because 5×5=255 \times 5 = 25.

Enter your number in the input field. Try 3636 and click Calculate to get 6.00006.0000. The result shows four decimal places for precision. Try 5050 to get 7.07117.0711, which is the approximate square root since 5050 isn't a perfect square. Try 22 to see 1.41421.4142, the famous square root of two.

Square roots only work with non-negative numbers in real mathematics. If you enter a negative number like −25-25 and click Calculate, you'll see an error message: "Error: Cannot calculate square root of negative number." This is mathematically correct—negative numbers don't have real square roots.

Perfect squares like 44, 99, 1616, 2525, 3636, 4949, 6464, 8181, and 100100 give whole number results. Non-perfect squares give decimal approximations. The calculator computes these accurately using JavaScript's Math.sqrt() function. Check the right sidebar for links to perfect squares tables showing common square roots.

Using Cube Root Mode

Select Cube Root to activate cube root mode. The display changes to show the ∛ symbol. Cube roots ask: "What number multiplied by itself twice equals my input?" For 273\sqrt[3]{27}, the answer is 33 because 3×3×3=273 \times 3 \times 3 = 27.

Enter a number and click Calculate. Try 88 to get 2.00002.0000 because 23=82^3 = 8. Try 6464 to get 4.00004.0000 because 43=644^3 = 64. Try 10001000 to get 10.000010.0000 because 103=100010^3 = 1000. These perfect cubes give exact whole number results.

Unlike square roots, cube roots work with negative numbers. Enter −8-8 and click Calculate to get −2.0000-2.0000. This works because (−2)×(−2)×(−2)=−8(-2) \times (-2) \times (-2) = -8. Try −27-27 to get −3.0000-3.0000 or −125-125 to get −5.0000-5.0000. Negative times negative times negative equals negative.

For non-perfect cubes, you'll see decimal approximations. Enter 1010 to get approximately 2.15442.1544. Enter 5050 to get approximately 3.68403.6840. The calculator uses JavaScript's Math.pow() function with an exponent of 1/31/3 to compute cube roots accurately. The right sidebar provides links to perfect cubes tables for reference.

Using Nth Root Mode

Click Nth Root to activate the most flexible mode. Two input fields appear: a small box on the left for the n value, and the main box for your number. The nth root asks: "What number raised to the nth power equals my input?" For the 4th root of 1616, the answer is 22 because 24=162^4 = 16.

Start by entering the n value in the small left box. Type 44 to calculate fourth roots, 55 for fifth roots, 66 for sixth roots, and so on. The n value must be positive and greater than zero. Hover over the question mark tooltip to see: "Enter the value of n for the nth root."

After setting n, enter your main number in the larger input field. For example, set n to 44 and enter 8181. Click Calculate to get 3.00003.0000 because 34=813^4 = 81. Set n to 55 and enter 3232 to get 2.00002.0000 because 25=322^5 = 32.

For even roots (n = 22, 44, 66, etc.), negative numbers cause errors just like square roots. Try n = 44 with input −16-16—you'll see "Error: Cannot calculate even root of negative number." For odd roots (n = 33, 55, 77, etc.), negative numbers work fine. Set n = 55 and enter −32-32 to get −2.0000-2.0000 successfully.

Reading Results and Understanding Output

After clicking Calculate, your result appears in the rightmost box of the calculation display. The result shows four decimal places for precision: 2.00002.0000, 3.16233.1623, or 1.71001.7100. Whole number results like 55 display as 5.00005.0000 to maintain consistent formatting.

Results appear in green text when the calculation succeeds. Perfect roots show clean whole numbers: 49=7.0000\sqrt{49} = 7.0000, 1253=5.0000\sqrt[3]{125} = 5.0000, or 2564=4.0000\sqrt[4]{256} = 4.0000. Imperfect roots show decimal approximations: 50=7.0711\sqrt{50} = 7.0711, 103=2.1544\sqrt[3]{10} = 2.1544.

Error messages appear in red text when calculations fail. You might see "Error: Invalid input" if you leave fields empty or enter non-numeric characters. "Error: Cannot calculate square root of negative number" appears for negative inputs in square root mode. "Error: Cannot calculate even root of negative number" appears when using even nth roots with negatives.

The error "Error: Invalid nth root" means the n value is zero, negative, or not a number. Make sure n is a positive number greater than zero. If you see "Error: Calculation failed," try resetting and re-entering your values. The result box displays "—" (em dash) when no calculation has been performed yet.

Using the Explanation Panel

The right sidebar contains an explanation panel that updates based on your selected root type. At the top, you'll see a blue badge labeled "EXPLANATION" indicating this section provides educational context. The panel remains visible as you scroll, staying in position to assist you.

The explanation text changes when you switch root types. For Square Root, you'll read about values that multiply by themselves once. For Cube Root, the text explains multiplication by itself twice. For Nth Root, you get a general explanation about n-1 multiplications.

Each explanation includes a concrete example with actual numbers. Square root shows: "the square root of 2525 is 55, because 5×5=255 \times 5 = 25." Cube root shows: "the cube root of 2727 is 33, because 3×3×3=273 \times 3 \times 3 = 27." These examples make the concept tangible.

Below the explanation text, a "Learn more:" section appears with blue arrow links. These connect to related resources like perfect squares tables, perfect cubes tables, and visualizers. Click these links to access deeper educational content. The panel's sticky positioning keeps it visible while you work with the calculator, providing constant reference.

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. For comprehensive root theory and notation, see radical expressions and root properties.

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. For comprehensive coverage of root types, see root classifications and radical 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}. For detailed root properties and simplification techniques, see radical simplification and root operations.

Related Calculators and Mathematical Tools

Exponent Calculator - Compute powers and exponents, the inverse operation of roots. If roots ask "what base gives this result," exponents compute the result directly. Use both calculators together to understand the relationship between powers and roots.

Logarithm Calculator - Find exponents when you know the base and result. Logarithms, exponents, and roots form a connected triangle of inverse operations in mathematics.

Fraction Calculator - Simplify radical expressions that produce fractions. Many root calculations yield fractional results, and this calculator helps reduce them to lowest terms.

Complex Numbers Calculator - Handle square roots of negative numbers by working with imaginary numbers. While the root calculator rejects negative square roots, complex numbers extend mathematics to include them.

Perfect Squares Table - Reference chart showing squares and square roots of common integers. Quickly check if a number is a perfect square.

Perfect Cubes Table - Reference chart showing cubes and cube roots of common integers. Identify perfect cubes at a glance.

For deeper theoretical understanding, explore radical expressions, rational exponents, root simplification, irrational numbers, and nth root functions.