Root Calculator

What a root means

A root reverses a power. The square root of 81 asks which nonnegative number multiplied by itself gives 81, so the principal square root is 9. The number inside is the radicand and the small number naming the root is the index. If no index is written, mathematics means 2. A cube root uses index 3, so the cube root of 125 is 5 because 5 x 5 x 5 = 125.

This calculator shows the principal real root, exponent form, and a power check. The radical expression has one principal square root, even though the equation x^2 = 81 has two solutions. Keeping that distinction clear prevents a common sign mistake. A root is useful in algebra, geometry, science, and any situation where a measurement has been raised to a power.

In practical work, the index should match the dimension you are recovering. Area problems often use a square root, volume problems often use a cube root, and algebra problems may use a fourth or higher root. Naming the index in your written work makes the operation auditable for another reader.

A root can also be estimated by finding nearby perfect powers. Since 7^2 = 49 and 8^2 = 64, the square root of 50 must be just above 7. That quick estimate is a useful check on a calculator result.

Worked examples

A square with an area of 144 square feet has a side of sqrt(144) = 12 feet, since 12 x 12 = 144. A cube with a volume of 343 cubic inches has an edge of 343^(1/3) = 7 inches, since 7^3 = 343. A fourth root reverses a fourth power: the fourth root of 625 is 5.

For 50, sqrt(50) is exact and 7.071068 is a decimal approximation. Keep more digits during a longer calculation and round at the end. The check is a practical way to see how close a displayed approximation comes to the original value.

For hand calculations, bracket the entire expression before taking a root. The square root of 36 + 64 is not the same as the square root of 36 plus 64. In a formula, write sqrt(36 + 64) when the sum is the radicand. Parentheses keep the intended order visible and prevent an otherwise correct root operation from being applied to only one term.

Roots also appear in rates and averages. If a quantity grows by a factor of 16 over 4 equal power steps, the fourth root of 16 is 2, so each step doubles the quantity. That interpretation is different from dividing 16 by 4, which gives 4. The index tells you how many repeated multiplications are being reversed.

Negative inputs

Odd and even roots behave differently with negative inputs. The cube root of -27 is -3 because (-3)^3 = -27. The fifth root of -32 is -2 because (-2)^5 = -32. Odd powers preserve the sign, so their roots can return a negative real value. This is useful when an equation models direction or a signed measurement.

An even root of a negative number is not real. No real number squared equals -9 because every real square is zero or positive. The widget gives a clear message instead of a misleading decimal. Complex roots need a different convention and are outside this page. See Wolfram MathWorld's principal square root reference for the difference between a principal radical and all solutions of an equation.

Nth roots and exponents

The general rule is the nth root of a equals a^(1/n). The fourth root of 16 is 16^(1/4) = 2, and the sixth root of 64 is 2 because 2^6 = 64. Enter a positive whole index. Zero, negative values, and fractions are not valid indexes in this calculator.

Exponent notation helps when a root appears inside a longer formula. The fourth root of 81 can be written 81^0.25, and the result is 3. OpenStax explains rational exponents and why the reciprocal index reverses the power. For a non-perfect root, the display is approximate. Raising the displayed answer back to its index should return approximately the starting number.

When the radicand is a perfect power, the result is often an integer. When it is not, the decimal approximation depends on the number of digits displayed. The exponent form is still exact as a mathematical expression, even when the decimal is shortened.

For an index of 2, the calculator uses a square root. For an index of 3 and a negative radicand, it preserves the negative sign. For higher odd indexes, the same signed behavior applies. For higher even indexes, the input must be nonnegative for a real result.

Measurement and geometry

Roots appear whenever a dimension is hidden inside a power. A square garden with 400 square feet has a side of sqrt(400) = 20 feet. A cube-shaped box with 1,000 cubic inches has an edge of the cube root of 1,000 = 10 inches. The units work because the dimension carries the same exponent as the measurement.

For a right triangle, c = sqrt(a^2 + b^2). With legs of 6 and 8, c = sqrt(36 + 64) = sqrt(100) = 10. The NIST SI units reference is a useful reminder to keep units consistent before calculating. Decide what the input represents before taking a root. If an area is 50 square meters, sqrt(50) gives a length in meters. A plain count has no physical unit after its square root.

When comparing two answers, compare their checks rather than only their displayed decimals. A root rounded to 6 places may produce a check that differs in the seventh or eighth place. That is expected. A large mismatch usually means the index was entered incorrectly, the radicand was mistyped, or a negative input was treated as an even root.

For geometry, keep the measurement system consistent. If one side is in inches and another is in feet, convert before applying the Pythagorean theorem. A root cannot repair mixed units.

The calculator performs the numerical operation, while the person using it remains responsible for the meaning and units of the inputs.

Rounding in measurements

Choose precision after considering the measurement. Keep extra digits during intermediate steps, then round the final answer to match input quality. Financial and scientific reports should state the unit and the selected rule.

How to use the result

Enter the radicand, choose the index, and read the principal real root. Use index 2 for square roots, 3 for cube roots, or another positive whole index for an nth root. The widget shows exponent form and a power check. Empty fields return a clear message.

If sqrt(50) is displayed as 7.0710678119, squaring that display gives approximately 50.0000000001. The small difference comes from display rounding. Keep an exact radical when exactness matters. Write the unit beside the answer and state whether you need the principal root or every solution of an equation. See MathWorld's square root reference. All calculations happen in your browser, and no entered values are uploaded.