Radical Calculator
Our free arithmetic calculator solves radical problems. Get worked examples, visual aids, and downloadable results. Get results you can export or share.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Radical Calculator
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Formula: nth_root(x) = x^(1/n); sqrt(a*b) = sqrt(a) * sqrt(b)
Worked example — sqrt(72) = 6*sqrt(2) = 8.48528
Formula
nth_root(x) = x^(1/n); sqrt(a*b) = sqrt(a) * sqrt(b)
The nth root of x equals x raised to the power 1/n. Radicals can be simplified by extracting perfect nth power factors. Product and quotient rules allow multiplication and division of same-index radicals by combining radicands.
Worked Examples
Example 1: Simplifying the Square Root of 72
Problem:Simplify the square root of 72 and express in simplest radical form.
Solution:Prime factorize 72: 72 = 2^3 * 3^2 Group in pairs for square root: 2^3 = 2^2 * 2 (one pair of 2s extracted, one 2 remains) 3^2 = one pair of 3s extracted Extracted: 2 * 3 = 6 Remaining under radical: 2 Result: 6 * sqrt(2) Decimal: 6 * 1.41421 = 8.48528 Verification: (6 * sqrt(2))^2 = 36 * 2 = 72
Result:sqrt(72) = 6*sqrt(2) = 8.48528
Example 2: Multiplying Cube Roots
Problem:Multiply the cube root of 12 by the cube root of 18 and simplify.
Solution:cube_root(12) * cube_root(18) = cube_root(12 * 18) = cube_root(216) 216 = 6^3, so cube_root(216) = 6 Alternatively: 12 = 2^2 * 3 18 = 2 * 3^2 12 * 18 = 2^3 * 3^3 = (2*3)^3 = 6^3 Result: 6 (a perfect cube!)
Result:cube_root(12) * cube_root(18) = cube_root(216) = 6
Frequently Asked Questions
What is a radical in mathematics and how do you read radical notation?
A radical is a mathematical expression that represents the root of a number. The radical symbol consists of a check mark shape with a horizontal bar extending over the radicand (the number under the radical). The small number in the notch of the radical symbol is the index, indicating which root to take. A square root (index 2) is the most common and typically written without showing the index. A cube root has index 3, a fourth root has index 4, and so on. The expression denoting the nth root of x asks: what number raised to the nth power equals x? For example, the cube root of 27 is 3 because 3 cubed equals 27. Radicals are inverse operations to exponentiation, just as subtraction is the inverse of addition and division is the inverse of multiplication.
How do you simplify radicals and what does simplified form mean?
Simplifying a radical means extracting all possible perfect power factors from under the radical sign. For a square root, find the largest perfect square that divides the radicand. The square root of 72 simplifies to 6 times the square root of 2, because 72 = 36 * 2 and the square root of 36 is 6. For cube roots, extract perfect cube factors: the cube root of 54 = the cube root of 27 * 2 = 3 times the cube root of 2. The general process involves prime factorization: factor the radicand into primes, group the factors by the root index, and extract complete groups. A radical is fully simplified when the radicand has no perfect power factors (other than 1) matching the index, there are no fractions under the radical, and there are no radicals in the denominator. This standard form makes it easier to compare, add, and multiply radical expressions.
What is the relationship between radicals and rational exponents?
Radicals and rational exponents are equivalent notations for the same mathematical concept. The nth root of x equals x raised to the power 1/n. More generally, x raised to the power m/n equals the nth root of x raised to the mth power, which also equals the nth root of x^m. For example, 8 to the power 2/3 equals the cube root of 8 squared, which equals 2 squared, which equals 4. Rational exponent notation is often preferred in algebra and calculus because it follows the same rules as integer exponents: x^(a/b) * x^(c/d) = x^(a/b + c/d), and (x^(a/b))^(c/d) = x^(ac/bd). This equivalence was formalized in the 16th century and is now fundamental to mathematical notation. Computer programming languages typically use the exponent form (Math.pow or ** operator) rather than radical notation.
How do you multiply and divide radicals?
Multiplying radicals with the same index is straightforward: the product of the nth root of a and the nth root of b equals the nth root of a times b. For example, the square root of 3 times the square root of 12 equals the square root of 36, which equals 6. After multiplying, always simplify the result. Division works similarly: the nth root of a divided by the nth root of b equals the nth root of a/b. The square root of 50 divided by the square root of 2 equals the square root of 25, which equals 5. For radicals with different indices, convert to rational exponents first: the square root of 2 times the cube root of 3 becomes 2^(1/2) * 3^(1/3) = 2^(3/6) * 3^(2/6) = the sixth root of (2^3 * 3^2) = the sixth root of 72. This common-index technique allows multiplication of any two radicals.
How do you add and subtract radicals?
Adding and subtracting radicals requires like radicals, which are radicals with the same index and the same radicand, analogous to like terms in algebra. You can add 3 times the square root of 5 and 7 times the square root of 5 to get 10 times the square root of 5, but you cannot directly combine the square root of 2 and the square root of 3. Before concluding that radicals cannot be combined, simplify each term first. For example, the square root of 12 plus the square root of 27 appears uncombineable, but simplifying gives 2 times the square root of 3 plus 3 times the square root of 3, which equals 5 times the square root of 3. This simplification step is crucial and often reveals like radicals hidden in seemingly different expressions. When radicals truly have different radicands after simplification, the sum can only be expressed as a decimal approximation.
What does it mean to rationalize the denominator and why do we do it?
Rationalizing the denominator is the process of eliminating radical expressions from the denominator of a fraction. For a simple radical denominator like 1 divided by the square root of 3, multiply both numerator and denominator by the square root of 3 to get the square root of 3 over 3. For denominators with two terms involving radicals, like 1 divided by (the square root of 5 plus the square root of 2), multiply by the conjugate (the square root of 5 minus the square root of 2) to eliminate the radicals using the difference of squares pattern. Historically, rationalization was practical because computing with radicals in the denominator was difficult before calculators. Today, it remains a standard convention in mathematics because rationalized form is considered simpler and makes it easier to compare expressions and identify equivalent forms. In some contexts like integration, rationalization is a necessary algebraic step.
What are imaginary numbers and how do they relate to even-index radicals?
When the radicand is negative and the index is even, the result involves imaginary numbers. The square root of negative 1 is defined as i, the imaginary unit. Therefore, the square root of any negative number can be expressed using i: the square root of -9 equals 3i, the square root of -2 equals i times the square root of 2. For even indices higher than 2, similar principles apply: the fourth root of -16 involves the fourth root of -1, which has complex values. Odd-index radicals of negative numbers are real: the cube root of -27 is -3, since (-3)^3 = -27. This distinction between even and odd indices reflects a fundamental asymmetry in the real number system. Complex numbers, which combine real and imaginary parts as a + bi, extend the real numbers to provide roots for all polynomial equations, a result known as the Fundamental Theorem of Algebra.
What are conjugate radicals and how are they used?
Conjugate radicals are pairs of expressions that differ only in the sign between terms, such as (a + the square root of b) and (a - the square root of b). When multiplied together, they produce a rational number using the difference of squares identity: (a + the square root of b)(a - the square root of b) = a squared minus b. This property makes conjugates essential for rationalizing denominators, solving radical equations, and simplifying expressions. In the quadratic formula, the two solutions are conjugate radicals: (-b + the square root of (b^2 - 4ac))/(2a) and (-b - the square root of (b^2 - 4ac))/(2a). The product of conjugate radical solutions equals c/a (by Vieta formulas), which is always rational when the coefficients are rational. Conjugates also appear in complex number theory, where the complex conjugate of a + bi is a - bi.
How do nested radicals work and can they be simplified?
Nested radicals are radical expressions inside other radical expressions, like the square root of (2 + the square root of 3). Some nested radicals can be denested into simpler forms. The classic example is the square root of (5 + 2 times the square root of 6), which equals the square root of 2 plus the square root of 3. To denest the square root of (a + b times the square root of c), look for values p and q such that p + q = a and p * q = b^2 * c / 4. If found, the nested radical equals the square root of p plus the square root of q. Ramanujan discovered many remarkable nested radical identities, including infinite nested radicals that converge to specific values. Not all nested radicals can be denested using radicals of lower depth. Determining whether a nested radical can be simplified is related to Galois theory and the solvability of polynomial equations.
Where are radicals used in science, engineering, and everyday life?
Radicals appear throughout practical applications. The Pythagorean theorem uses square roots to find distances: the diagonal of a rectangle with sides a and b is the square root of (a^2 + b^2). In physics, the period of a pendulum involves the square root of its length divided by gravitational acceleration. Standard deviation in statistics is the square root of variance. The quadratic formula, fundamental to algebra and engineering, requires square roots. Electrical engineering uses square roots in calculating RMS (root mean square) voltage. In computer graphics, distance calculations and vector normalization use square roots constantly. Cube roots appear in volume calculations and in determining the side length of a cube with a given volume. Financial mathematics uses nth roots to calculate annualized returns: the annualized return over n years equals the nth root of the cumulative return factor minus 1.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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