Simplify Radicals
Simplify any square root to its simplest radical form (a√b) instantly. Free online simplify radicals calculator extracts perfect squares using prime factorization and shows both exact form and decimal value.
About This Calculator
Our simplify radicals calculator helps students, teachers, and math enthusiasts quickly rewrite any square root in its simplest radical form. Instead of just giving you the decimal approximation, this tool performs prime factorization on your number, extracts all perfect square factors, and presents the result as asqrtb -- the exact simplified radical expression used in algebra, geometry, and advanced mathematics.
The calculator works by finding the prime factorization of the input number, grouping identical prime factors into pairs, and moving each complete pair outside the radical. For example, entering 72 produces the result 6sqrt2 because 72 factors as 2^3 x 3^2 = 2^2 x 2 x 3^2, and the pairs of 2s and 3s move outside as 2 x 3 = 6, leaving a single 2 inside. The tool also shows the decimal square root value alongside the simplified form, giving you both the exact expression and its approximate numeric value.
This calculator supports non-negative real numbers, perfect squares, and any integer. Perfect squares like 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 simplify to whole numbers. Squarefree numbers (prime numbers and products of distinct primes) like 2, 3, 5, 6, 7, 10, and 15 stay in simplest radical form. This makes it ideal for algebra homework, geometry problems, precalculus studies, and any situation where exact radical expressions are preferred over decimal approximations.
Regional Notes
Radical simplification is a universal mathematical operation with no regional variations. The prime factorization method used here follows standard arithmetic conventions taught worldwide, including in India (CBSE/ICSE), the United States (CCSS), and the United Kingdom (National Curriculum). The sqrt symbol and radical notation are mathematically identical across all regions.
Frequently Asked Questions
What does it mean to simplify a radical (square root)?
Simplifying a square root means rewriting it in the form asqrtb where no perfect square factors remain inside the radical. For example, sqrt72 simplifies to 6sqrt2 because 72 = 36 x 2 and sqrt36 = 6.
How do you simplify a square root step by step?
First, find the prime factorization of the number. Group identical prime factors into pairs. Each complete pair moves outside the radical as a single factor. Any remaining unpaired factors stay inside. For example, sqrt72 = sqrt(2^3 x 3^2) = sqrt(2^2 x 2 x 3^2) = 2 x 3 x sqrt2 = 6sqrt2.
Can you simplify the square root of a negative number?
Square roots of negative numbers are not real numbers; they belong to the realm of imaginary numbers (involving i, the imaginary unit). This calculator works with real numbers only. For negative inputs, consider using an imaginary numbers calculator.
What is the difference between simplifying a square root and finding its decimal value?
Simplifying expresses the square root in exact radical form (asqrtb) by extracting perfect squares, while the decimal value is an approximation. The simplified form is exact and preferred in algebra, while the decimal is useful for practical measurements. For example, sqrt72 is exactly 6sqrt2, but approximately 8.485.
Is every square root simplifiable?
Only numbers that contain perfect square factors can be simplified. If a number's prime factorization has no exponent of 2 or higher, the square root is already in simplest radical form, such as sqrt2, sqrt3, sqrt5, sqrt6, sqrt7, sqrt10, and other squarefree numbers.
How do you simplify the square root of a fraction?
Apply the square root separately to numerator and denominator: sqrt(a/b) = sqrta / sqrtb. Simplify each part individually. For example, sqrt(4/9) = sqrt4 / sqrt9 = 2/3. For fractions, both numerator and denominator must be perfect squares for a whole-number result.
What are perfect squares and why do they matter?
Perfect squares are numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 whose square roots are integers. They matter because they can be fully extracted from a radical, leaving no factors inside. Recognizing perfect squares helps simplify radicals quickly.
Is this simplify radicals calculator free to use?
Yes, all calculators on Calculy are completely free to use with no limits, registration, or hidden charges.