Radical Simplifier

Use this Radical Simplifier to enter values, adjust options, and review results in a compact responsive workspace.

Results are calculated automatically as you enter data.

Radical expression

Enter a square root with an optional coefficient, such as sqrt(72), 1/3sqrt(2), 3sqrt(50), or 5*sqrt(18). Coefficients may use up to 100 total digits across a fraction; exponent notation is not supported.

Exact simplified form

Decimal approximation

Perfect-square factor

Step hints

Disclaimer: All tools on this site are intended for educational and informational purposes only. Estimate only; not professional advice. Verify units, assumptions, precision, and method before relying on results. See our Terms of Use.

Assumptions and limitations

This Radical Simplifier is for general educational use. Results are estimates or mathematical outputs based on the inputs and stated assumptions, not professional engineering, safety, financial, legal, or academic advice. Check units, rounding, formulas, and limitations before using results for important work. For professional, safety-sensitive, construction, engineering, official, or high-cost uses, verify results with an appropriate method, current standards, required tolerances, and a qualified professional. Use of this calculator is subject to our Terms of Use.

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Related Calculators

What Is Simplifying Square-Root Radical Expressions?

Simplifying a square-root radical expression means rewriting it in an equivalent form that is easier to read, compare, and use in later algebra steps. The value does not change. Only the form changes.

For example, $\sqrt{72}$ can be rewritten as $6\sqrt{2}$. These two expressions are exactly equal, but $6\sqrt{2}$ is considered simpler because the number left under the square root has no perfect-square factor greater than $1$.

A square-root radical is usually written with the radical sign $\sqrt{\ }$. In calculator input, it is often typed as sqrt(...), such as sqrt(72) or 3sqrt(50).

Simplifying radicals matters because exact radical form often gives more information than a rounded decimal. A decimal approximation such as $8.4852813742$ is useful for estimating size, but $6\sqrt{2}$ preserves the exact value.

Why Simplifying Square Roots Matters

Simplified radicals are common in algebra, geometry, trigonometry, physics, and many word problems involving distance or length. They are especially useful when an answer is expected in exact form.

Simplifying square roots helps you:

Recognize equivalent answers, such as $\sqrt{72}$ and $6\sqrt{2}$.
Combine like radicals after simplification.
Keep exact answers instead of relying only on rounded decimals.
See the structure of a number by separating its perfect-square part from its square-free part.
Check whether a radical can be reduced any further.

For students, this skill also builds fluency with factors, perfect squares, and the product property of square roots.

Key Terms to Know

Square root: A number that produces the original number when multiplied by itself. For example, $5$ is a square root of $25$ because $5^2 = 25$.
Principal square root: The nonnegative square root shown by the radical sign. For example, $\sqrt{25} = 5$.
Radical expression: An expression that contains a radical sign, such as $\sqrt{18}$ or $4\sqrt{7}$.
Radicand: The number or expression under the radical sign. In $\sqrt{72}$, the radicand is $72$.
Coefficient: The number multiplying the radical. In $3\sqrt{50}$, the coefficient is $3$.
Perfect square: A number that is the square of an integer, such as $1, 4, 9, 16, 25, 36, 49,$ and $64$.
Square-free radicand: A radicand with no perfect-square factor greater than $1$. For example, $2$, $3$, $5$, $6$, and $7$ are square-free.
Exact form: A mathematically exact expression, such as $6\sqrt{2}$.
Decimal approximation: A rounded decimal version of the value, such as $8.4852813742$.

How Simplifying Square Roots Works

The main idea is to break the radicand into two factors: the largest perfect-square factor and the remaining factor. Then the square root of the perfect-square factor can move outside the radical.

The product property of square roots says that for nonnegative numbers $a$ and $b$:

$$ \sqrt{ab} = \sqrt{a}\times\sqrt{b} $$

For a radical with an outside coefficient, write the expression as:

$$ c\sqrt{n} $$

Where:

$c$ is the outside coefficient.
$n$ is the nonnegative whole-number radicand.
$s$ is the largest perfect-square factor of $n$.
$k = \sqrt{s}$.
$r = \frac{n}{s}$ is the remaining radicand.

Then:

$$ c\sqrt{n} = c\sqrt{s\times r} $$

Using the product property:

$$ c\sqrt{s\times r} = c\sqrt{s}\sqrt{r} $$

Since $s$ is a perfect square, $\sqrt{s}=k$, so:

$$ c\sqrt{n} = ck\sqrt{r} $$

If $r = 1$, the radical disappears and the result is a plain number:

$$ ck\sqrt{1} = ck $$

If the coefficient is $0$, or if the radicand is $0$, the whole expression simplifies to $0$.

Examples of Simplifying Square Roots in Practice

Example 1: Simplifying a Basic Radical

Simplify $\sqrt{72}$.

The largest perfect-square factor of $72$ is $36$:

$$ 72 = 36\times 2 $$

Rewrite the square root:

$$ \sqrt{72} = \sqrt{36\times 2} $$

Separate the factors:

$$ \sqrt{36\times 2} = \sqrt{36}\times\sqrt{2} $$

Simplify $\sqrt{36}$:

$$ \sqrt{36}\times\sqrt{2} = 6\sqrt{2} $$

So:

$$ \sqrt{72} = 6\sqrt{2} $$

Example 2: Simplifying with an Outside Coefficient

Simplify $3\sqrt{50}$.

The largest perfect-square factor of $50$ is $25$:

$$ 50 = 25\times 2 $$

Rewrite and simplify:

$$ 3\sqrt{50} = 3\sqrt{25\times 2} $$

$$ 3\sqrt{25\times 2} = 3\times\sqrt{25}\times\sqrt{2} $$

$$ 3\times 5\sqrt{2} = 15\sqrt{2} $$

So:

$$ 3\sqrt{50} = 15\sqrt{2} $$

The important step is multiplying the outside coefficient by the factor that comes out of the radical. The $3$ does not stay unchanged when $\sqrt{25}=5$ moves outside.

Example 3: Simplifying a Perfect Square

Simplify $\sqrt{144}$.

Since $144 = 12^2$, the square root is a whole number:

$$ \sqrt{144} = 12 $$

There is no radical left because the radicand itself is a perfect square.

Example 4: Simplifying with a Negative Coefficient

Simplify $-2\sqrt{45}$.

The largest perfect-square factor of $45$ is $9$:

$$ 45 = 9\times 5 $$

Then:

$$ -2\sqrt{45} = -2\sqrt{9\times 5} $$

$$ -2\sqrt{9\times 5} = -2\times 3\sqrt{5} $$

$$ -2\times 3\sqrt{5} = -6\sqrt{5} $$

So:

$$ -2\sqrt{45} = -6\sqrt{5} $$

The negative sign stays outside the radical. This is different from trying to take the square root of a negative radicand.

Example 5: A Radical That Is Already Simplified

Simplify $\sqrt{19}$.

The number $19$ has no perfect-square factor greater than $1$. That means it is already in simplified radical form:

$$ \sqrt{19} = \sqrt{19} $$

A radical does not need to become a whole number or a decimal to be simplified.

How to Interpret the Result

The exact simplified form is the most important result. It is mathematically equal to the original radical expression, but written with all removable perfect-square factors outside the square root.

The decimal approximation is useful for estimating the size of the value. For example, $6\sqrt{2}$ is approximately $8.4852813742$. However, a decimal may be rounded, so it should not be treated as more exact than the radical form.

The square-factor split shows why the simplification works. A split such as $36 \times 2$ means the radicand was separated into a perfect square and a leftover factor:

$$ \sqrt{72} = \sqrt{36\times 2} $$

If the split begins with $1$, the radicand has no perfect-square factor greater than $1$, so the radical is already simplified.

If the remaining radicand is $1$, the result becomes a plain number. For example:

$$ \sqrt{144} = \sqrt{144\times 1} = 12 $$

A simplified radical is not a score, so there is no “high” or “low” result in the way there might be for a measurement or percentage. A larger decimal value simply means the expression has a larger numerical value.

Common Mistakes and Misconceptions

Mistake 1: Forgetting the Outside Coefficient

When simplifying $3\sqrt{50}$, the $5$ from $\sqrt{25}$ must multiply the existing coefficient $3$:

$$ 3\sqrt{50} = 3\times 5\sqrt{2} = 15\sqrt{2} $$

Writing $3\sqrt{50}=5\sqrt{2}$ ignores the coefficient and changes the value.

Mistake 2: Splitting Addition Inside a Square Root

The product property works for multiplication, not addition. In general:

$$ \sqrt{a+b} \ne \sqrt{a}+\sqrt{b} $$

For example:

$$ \sqrt{9+16} = \sqrt{25}=5 $$

But:

$$ \sqrt{9}+\sqrt{16}=3+4=7 $$

The two results are not equal.

Mistake 3: Confusing Exact Form with Decimal Form

A simplified radical such as $6\sqrt{2}$ is exact. A decimal such as $8.4852813742$ is an approximation. Use the exact radical form when an exact answer is required.

Mistake 4: Expecting a Real Answer for a Negative Radicand

In real-number algebra, an even root such as a square root requires a nonnegative radicand. An expression like $\sqrt{-9}$ is not a real number. It belongs to complex-number work, not basic real radical simplification.

Mistake 5: Entering a Multi-Term Expression as One Radical

A single radical such as $\sqrt{72}$ can be simplified by removing perfect-square factors. A larger expression such as $\sqrt{8}+\sqrt{18}$ requires simplifying each radical separately and then combining like radicals if possible.

When to Use Simplifying Square Roots

Use square-root simplification when you need to:

Rewrite a radical in simplest exact form.
Check whether two radical expressions are equivalent.
Prepare radicals for addition or subtraction.
Simplify exact answers in algebra or geometry.
Convert a radical to a decimal estimate after finding its exact form.
Understand how a radicand factors into a perfect-square part and a leftover part.

This process is especially helpful when solving equations, working with right triangles, simplifying formulas, or checking homework answers that require exact form.

Limitations and Things to Keep in Mind

Simplifying a square root does not solve every kind of radical problem. It only rewrites a square-root expression in an equivalent simpler form.

This calculator works with a single real square-root expression that has an optional outside numeric coefficient and a nonnegative whole-number radicand. It is not meant for expressions with variables, nested radicals, multiple radicals, addition or subtraction between radicals, rationalizing denominators, cube roots, fourth roots, or general $n$th roots.

The radicand must be a whole number. Decimal and fraction values may be used as outside coefficients, but not as the number under the square root. An outside coefficient may contain at most 100 total digit characters across its numerator and denominator; signs, decimal points, and the fraction slash do not count. Exponent notation is not supported.

Negative radicands are not simplified because the calculator works in the real numbers. If you are studying complex numbers, a negative radicand requires different rules involving $i$, where $i^2=-1$.

The decimal approximation may be rounded to as many as $10$ decimal places, with unnecessary trailing zeros removed. For exact algebra work, use the exact simplified form rather than the decimal approximation.

Radicands are limited to nonnegative safe whole numbers up to $10,000,000,000$ so factorization remains prompt. The complete outside coefficient has at most 100 total digit characters across its numerator and denominator, excluding signs, decimal points, and the fraction slash; this includes leading and trailing decimal zeros and is checked before exact-integer construction. Exponent notation is not supported. Decimal approximations are separate from the exact result and can be unavailable when an extremely large coefficient cannot be represented as a finite JavaScript decimal.

How to Use This Calculator

Enter one square-root expression in a supported form, such as sqrt(72), sqrt72, 3sqrt(50), or 3*sqrt(50).
Include an outside coefficient if needed. The coefficient may be positive, negative, a decimal, or a fraction.
Keep the radicand as a nonnegative whole number.
Review the exact simplified form first.
Use the decimal approximation only when you need an estimated value.
Check the square-factor split to see which perfect-square factor was removed from the radicand.
Read the step-by-step explanation to follow the simplification process.
Use the example buttons for sample expressions, Copy to copy a valid result, or Clear to reset the input.

Frequently Asked Questions

What does simplest radical form mean?

For a square root, simplest radical form usually means the radicand has no perfect-square factor greater than $1$. For example, $\sqrt{72}$ is not simplified because $36$ is a perfect-square factor. The simplified form is $6\sqrt{2}$.

Why does $\sqrt{72}$ become $6\sqrt{2}$?

Because $72$ can be factored as $36\times 2$, and $36$ is a perfect square. The square root of $36$ is $6$, so that part moves outside the radical while $2$ stays inside.

Can a radical be simplified if the radicand is prime?

Usually not, unless there is a coefficient outside that can be simplified separately. A prime radicand such as $19$ has no perfect-square factor greater than $1$, so $\sqrt{19}$ is already simplified.

What is the difference between exact form and decimal approximation?

Exact form preserves the value without rounding, such as $15\sqrt{2}$. A decimal approximation gives a rounded numeric estimate, such as $21.2132034356$. Use exact form for algebraic work and decimals for estimation.

Can I simplify a negative radicand?

Not as a real square root. In real-number algebra, $\sqrt{-9}$ is not a real number. Complex-number simplification uses different notation and is outside the scope of this calculator.

Can this calculator simplify expressions like $\sqrt{8}+\sqrt{18}$?

No. That is a multi-term radical expression. You can simplify each radical separately, but combining the terms requires additional algebra beyond simplifying one square root.

Sources and References

Books

Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis. Elementary Algebra 2e. OpenStax, 2020. Chapter 9, Sections 9.1 “Simplify and Use Square Roots” and 9.2 “Simplify Square Roots.” https://openstax.org/books/elementary-algebra-2e/pages/9-1-simplify-and-use-square-roots and https://openstax.org/books/elementary-algebra-2e/pages/9-2-simplify-square-roots
Lynn Marecek and Andrea Honeycutt Mathis. Intermediate Algebra 2e. OpenStax, 2020. Chapter 8, Section 8.2 “Simplify Radical Expressions.” https://openstax.org/books/intermediate-algebra-2e/pages/8-2-simplify-radical-expressions
Jay Abramson. College Algebra 2e. OpenStax, 2021. Chapter 1, Section 1.3 “Radicals and Rational Exponents.” https://openstax.org/books/college-algebra-2e/pages/1-3-radicals-and-rational-exponents