Radical Simplifier, Explained
After reading this you will be able to reduce any square root or higher root to its simplest exact form, rationalize a denominator by hand, and check the calculator's answer with a single multiplication.
What a radical simplifier does
A radical is a root written with the sign \sqrt{\ }. Simplifying a radical means rewriting it so the number under the root has no perfect-square factor left, so no root sits in a denominator, and so any nested roots are flattened when that is possible. The value never changes. Only the form does.
The classic example is sqrt(72). The number 72 factors as 72 = 36 \times 2, and 36 is a perfect square, so \sqrt{72} = \sqrt{36}\,\sqrt{2} = 6\sqrt{2}. Both forms equal 8.485 to four figures, but 6√2 is exact and tells you the structure at a glance.
Exact and decimal are two views of the same number. 6√2 and 8.485281... agree to every digit. The exact form is preferred in algebra because it keeps the \sqrt{2} visible, which matters when the result feeds into another calculation.
When to use it, and when not
Reach for exact simplification when you are doing algebra, geometry with right triangles, or trigonometry. The side of a square with area 72 is 6√2, and a diagonal of a unit square is √2. These are the shapes of answers your teacher and your textbook expect.
Skip it when you only need a measurement. If you are cutting a board and want the length in centimeters, the decimal 8.49 is more useful than 6√2. Simplification is about form, not about getting a number you can act on physically.
The tool also handles cube roots and higher roots, and it rationalizes denominators such as 1/(1 + sqrt(2)), which is the demo example worked below.
The rule for a single root
Every simplification of a single square root rests on one identity.
Here a and b are non-negative numbers. You pull out the largest perfect-square factor. To simplify \sqrt{n}, write n = s^2 \cdot m where s^2 is the largest square dividing n, then \sqrt{n} = s\sqrt{m}.
The same idea works for an nth root: pull out any factor that is a perfect nth power.
For example \sqrt[3]{54} = \sqrt[3]{27 \times 2} = 3\sqrt[3]{2}, since 27 is 3^3. Check: 3\sqrt[3]{2} \approx 3 \times 1.2599 = 3.780, and 54^{1/3} \approx 3.780 as well.
Finding the largest square factor
The reliable method is prime factorization. Break the number into primes, then take out each prime in pairs (for a square root) or in triples (for a cube root).
Take 72. Its prime factorization is 2^3 \times 3^2. Group into pairs: one pair of 2s, one pair of 3s, and a single 2 left over. Each pair leaves the room as one copy of that prime, so you take out 2 \times 3 = 6 and leave the lone 2 inside. That gives 6√2.
| Number | Prime factors | Pairs pulled out | Simplified |
|---|---|---|---|
| 72 | 2^3 \cdot 3^2 | 2, 3 | 6\sqrt{2} |
| 50 | 2 \cdot 5^2 | 5 | 5\sqrt{2} |
| 200 | 2^3 \cdot 5^2 | 2, 5 | 10\sqrt{2} |
| 45 | 3^2 \cdot 5 | 3 | 3\sqrt{5} |
| 17 | 17 | none | \sqrt{17} |
When there are no repeated primes, as with 17, the root is already in simplest form.
Rationalizing a denominator
A denominator with a root is considered unsimplified. To remove a single root, multiply top and bottom by that root. To remove a sum like 1 + \sqrt{2}, multiply by its conjugate, which is the same expression with the sign flipped: 1 - \sqrt{2}. The conjugate works because of the difference of squares.
When a and b involve square roots, squaring them removes the root. That is the whole trick.
Worked example: 1/(1 + √2)
This is the calculator's demo. Enter 1/(1 + sqrt(2)) and press Calculate.
- The denominator is 1 + \sqrt{2}. Its conjugate is 1 - \sqrt{2}.
- Multiply numerator and denominator by that conjugate: \dfrac{1}{1+\sqrt{2}} \cdot \dfrac{1-\sqrt{2}}{1-\sqrt{2}}.
- The new denominator is (1+\sqrt{2})(1-\sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1.
- The new numerator is 1 \cdot (1 - \sqrt{2}) = 1 - \sqrt{2}.
- Divide: \dfrac{1-\sqrt{2}}{-1} = \sqrt{2} - 1.
So \dfrac{1}{1+\sqrt{2}} = \sqrt{2} - 1. Check numerically: the left side is 1 / 2.4142 = 0.4142, and the right side is 1.4142 - 1 = 0.4142. They match, and the root is gone from the denominator.
Reading the calculator's output
The output is one exact expression. Read it in three parts: the integer factor pulled out, the reduced radicand still under the root, and the index if it is not a square root. In 6√2 the 6 is the pulled-out factor and 2 is the reduced radicand. When rationalizing, expect the root to disappear from the bottom of the fraction, as in the demo where 1/(1 + √2) became √2 − 1 with no fraction left at all.
To confirm any result, compute both forms as decimals and compare. If they agree to the digits shown, the simplification is valid. This chart shows several roots and their simplified equivalents landing on the same decimal value.
Common mistakes
Three errors show up again and again.
- Splitting a sum under the root
- \sqrt{a+b} is not \sqrt{a} + \sqrt{b}. Test it: \sqrt{9+16} = \sqrt{25} = 5, but \sqrt{9} + \sqrt{16} = 3 + 4 = 7. The product rule applies only to multiplication.
- Stopping too early
- Writing \sqrt{72} = 2\sqrt{18} is correct but not simplified, since 18 still contains the square 9. Keep going until the radicand is square-free: 2\sqrt{18} = 2 \cdot 3\sqrt{2} = 6\sqrt{2}.
- Losing a sign
- For real numbers \sqrt{x^2} = |x|, not x. If x = -5, then \sqrt{(-5)^2} = \sqrt{25} = 5, the absolute value, not -5.
A square root of a negative number is not real. \sqrt{-4} has no real value; it equals 2i in the complex numbers. Use the Complex number calculator when the radicand is negative.
Related tools
Simplified radicals usually appear as answers from other work. Right triangle and distance problems produce them, and quadratics produce them through the square root in the formula. The Quadratic equation solver reports exact roots such as \frac{-3 + \sqrt{5}}{2}, which are radicals in disguise. For roots of higher-degree polynomials, use the Equation solver. When you need logarithms or an nth root as a decimal, the Exponent and logarithm calculator handles those, and the Partial fraction decomposition tool splits rational expressions before you simplify their pieces.
Frequently asked questions
Why is 6√2 called simpler than √72?
Because 2 is square-free, so no further extraction is possible, and the factor 6 shows the size of the number at a glance. Both equal 8.485, but 6√2 reveals the structure and combines cleanly with other terms, for example 6\sqrt{2} + 4\sqrt{2} = 10\sqrt{2}.
How do I simplify a cube root?
Pull out perfect cubes instead of perfect squares. For \sqrt[3]{54}, factor 54 as 27 \times 2 = 3^3 \times 2, take the 3 out, and get 3\sqrt[3]{2} \approx 3.780.
What does rationalizing the denominator accomplish?
It moves every root to the numerator, giving an integer or polynomial denominator. This makes fractions easier to add and compare. 1/(1 + √2) becomes √2 − 1, which you can add to another number without a root sitting underneath.
Can every nested root be denested?
No. \sqrt{3 + 2\sqrt{2}} denests to 1 + \sqrt{2} because (1+\sqrt{2})^2 = 3 + 2\sqrt{2}, but most nested roots such as \sqrt{1 + \sqrt{2}} have no simpler form. The calculator denests only when a valid form exists.
Is √2 − 1 really equal to 1/(1 + √2)?
Yes. Multiply (\sqrt{2}-1)(\sqrt{2}+1) = 2 - 1 = 1, which shows the two are reciprocals of \sqrt{2}+1 from opposite sides. Numerically both equal 0.4142.
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