About the Simplify Radical Expressions Calculator
Once single roots are in simplest form, the next step in algebra is working with expressions that contain several of them. You add and subtract radicals much as you combine like terms in algebra, and you multiply them by multiplying what is under the roots. The trick is that terms that look unlike, such as √12 and √27, can turn out to be like radicals once each is simplified.
This simplify radical expressions calculator takes an expression with square roots joined by plus, minus and multiplication, such as 3√12 + 5√27 − √48 or 2√6 × 3√2. It simplifies each term, combines like radicals and returns an exact answer with a decimal check, showing the simplified form of every term along the way.
How to Use the Simplify Radical Expressions Calculator
Type the expression. Write square roots with √ or sqrt, as in √12 or sqrt(12).
Use + and − between terms, and ×, ·, ***** or a space to multiply within a term.
Put any number in front of a root as its coefficient, including fractions such as 1/2√8 and decimals.
The Formulas
product rule: √a × √b = √(ab)
simplify a term: c√(k² × r) = ck√r, with r square-free
like radicals: p√r + q√r = (p + q)√r
unlike radicals: p√r + q√s stays as it is when r ≠ sStep-by-Step Example
Simplify 3√12 + 5√27 − √48.
3√12 = 3 × √(4 × 3) = 3 × 2√3 = 6√3
5√27 = 5 × √(9 × 3) = 5 × 3√3 = 15√3
√48 = √(16 × 3) = 4√3
6√3 + 15√3 − 4√3 = (6 + 15 − 4)√3 = 17√3 ≈ 29.4449Simplify 2√6 × 3√2. Multiply the coefficients, 2 × 3 = 6, and the radicands, 6 × 2 = 12, to get 6√12. Then √12 = 2√3, so the answer is 12√3 ≈ 20.7846.
Why Only Like Radicals Combine
Adding radicals works just like adding algebraic terms. 6√3 + 15√3 is six of something plus fifteen of the same thing, so it is 21 of it, exactly as 6x + 15x = 21x. But √2 and √3 are different irrational numbers with no common unit, so √2 + √3 cannot be written as a single radical. In particular it is not √5: √2 + √3 ≈ 3.146 while √5 ≈ 2.236.
That is why simplifying each term first is essential. √12, √27 and √48 look unrelated, but each contains the factor 3 under a perfect square, so all three become multiples of √3 and combine into one term.
Multiplying and Dividing Radicals
For non-negative numbers, the product of square roots is the square root of the product: √a × √b = √(ab). This lets you multiply radicals and then simplify. Sometimes the product is a perfect square and the root disappears: √12 × √3 = √36 = 6. Coefficients outside the roots multiply separately, then join whatever comes out of the root.
Division works the same way, √a ÷ √b = √(a/b), and a simplified answer should not leave a root in the denominator. For quotients, and for cube roots or higher roots, use the simplest radical form calculator, which rationalises fractions under any root.
Mixing Whole Numbers and Radicals
Whole numbers are radicals with nothing left under the root, so they combine with each other but not with radical terms. The expression √8 + √18 − √2 + 5 simplifies to 2√2 + 3√2 − √2 + 5 = 5 + 4√2. The calculator lists the whole-number part first, then each group of like radicals in increasing order of the number under the root.
If every radical cancels, the answer is a whole number or zero. √8 − 2√2 = 2√2 − 2√2 = 0, which is a quick way to check whether two radical expressions are equal.
Where These Expressions Appear
Radical expressions appear when solving quadratic equations, finding exact side lengths with the Pythagorean theorem, working with special right triangles and simplifying trigonometric values. The perimeter of a shape with sides √12, √27 and √48 is a direct example: 2√3 + 3√3 + 4√3 = 9√3. Exam questions often ask for answers in simplest radical form, so combining like radicals correctly matters for full marks.
Conjugates and Binomials With Radicals
Expressions such as (2 + √3)(2 − √3) are products of conjugates. Multiplying out gives 4 − 2√3 + 2√3 − 3 = 1: the radical terms cancel, which is why conjugates are used to rationalise denominators like 1/(2 + √3). Squaring a binomial with a radical works the same way as any binomial: (1 + √2)² = 1 + 2√2 + 2 = 3 + 2√2. These products need brackets and are best done by hand or with the polynomial multiplication calculator; this calculator handles sums of terms, each of which may be a product of numbers and roots.
Checking With Decimals
The decimal value is a useful check on hand working. If you simplify an expression to 17√3, the original and the answer should both come to about 29.445. A different decimal means a term was simplified incorrectly or a sign was lost.
Understanding Your Result
The headline gives the simplified expression in exact form.
The decimal line gives its value as a check.
The like radicals line says whether everything combined into one term or how many unlike terms remain.
The steps show each term's simplified form before the like terms are combined.
When Should You Use This Calculator?
Use it to add, subtract or multiply square roots in algebra homework.
Use it to check that two radical expressions are equal.
Use it to see which terms in a long expression are like radicals.
Common Mistakes
Adding the numbers under the roots. √2 + √3 is not √5.
Combining before simplifying. Simplify √12 and √27 first; they are both multiples of √3.
Forgetting coefficients when multiplying. 2√6 × 3√2 multiplies 2 by 3 as well.
Stopping too early. 6√12 is not simplest; it is 12√3.