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Simplify Radicals Calculator

Select the operation and provide the required entities. The calculator will simplify the radicals accordingly, with the steps shown.

basic radical

a (Optional):

c (Optional):

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Use this free online simplify radicals calculator to quickly convert any number or expression into its simplest radical form.

Scroll down to learn more about radicals and the mathematics behind simplifying them.

What Is a Radical?

A radical is represented by the symbol $ \sqrt{} $, which can indicate a square root or an nth root of a number or expression.

simplify radicals

Mathematically, a radical can be written as:

$$ y \cdot \sqrt[n]{x} $$

Where:

$ \sqrt{} $ is the radical symbol
x is the radicand (the number under the radical)
n indicates the nth root

With this free simplify radicals calculator, simplifying radical expressions has never been easier or faster.

For decimal exponents, you can also use our decimal calculator to handle fractional powers.

Operations on Radicals

Addition

Two radicals can only be added if their radicands and root orders are the same. Otherwise, simplify first before adding.

$$ a \sqrt[n]{b} + c \sqrt[m]{d} $$

$$ n = m, \quad b = d $$

Subtraction

The rules for subtraction are identical to addition, except a negative sign is used.

$$ a \sqrt[n]{b} - c \sqrt[m]{d} $$

$$ n = m, \quad b = d $$

Multiplication

To multiply radicals of the same order, multiply the radicands together:

$$ a \sqrt[n]{b} \cdot c \sqrt[n]{d} = (a \cdot c) \sqrt[n]{b \cdot d} $$

If the radicals have different roots, multiplication cannot be directly performed.

Division

Dividing radicals of the same root order involves canceling out common factors and simplifying:

$$ \frac{a \sqrt[n]{b}}{c \sqrt[n]{d}} $$

Our free calculator can simplify division problems instantly.

Rules to Simplify Radicals

The root order n must be ≥ 2
The radicand must be non-negative

Square Roots

Square roots are radicals with order 2. They are the inverse of squaring a number.

Example:

$ \sqrt{25} = 5 $ because $ 5^2 = 25 $

For larger numbers, our simplify radicals calculator can compute the square root in seconds.

How to Simplify Radical Expressions

Example 1: Addition

Simplify:

$$ \sqrt{75} + \sqrt{12} $$

Step 1: Factor radicands:

$$ \sqrt{75} + \sqrt{12} = \sqrt{25 \cdot 3} + \sqrt{4 \cdot 3} $$

Step 2: Take out perfect squares:

$$ 5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3} $$

Example 2: Multiplication

Simplify:

$$ \sqrt{18} \cdot \sqrt{14} $$

Step 1: Factor perfect squares:

$$ \sqrt{9 \cdot 2} \cdot \sqrt{14} = 3\sqrt{2} \cdot \sqrt{14} = 3\sqrt{28} $$

Step 2: Simplify further:

$$ 3\sqrt{4 \cdot 7} = 3 \cdot 2 \sqrt{7} = 6\sqrt{7} $$

Example 3: Simplifying a Number

Simplify 228:

$$ 228 = 2 \cdot 114 = 2 \sqrt{57} $$

How the Calculator Works

Input:

Select the operation type
Enter numbers in the appropriate fields
Click “Calculate”

Output:

Simplest radical form
Notification if the radical cannot be simplified

FAQs

Square root of 288 in radical form?

$$ \sqrt{288} = 12\sqrt{2} $$

Can you reverse a radical?

Yes, by finding the number which, when raised to the root’s power, returns the radicand.

Square root of 0?

Can roots be negative?

Yes, negative radicands yield complex numbers: $$ \sqrt{-4} = 2i $$

2 cubed?

$$ 2^3 = 2 \cdot 2 \cdot 2 = 8 $$

Why is 1729 special?

It is the sum of two sets of cubes in two distinct ways.

Radicals vs. radicands?

Radicals with the same root and radicand are called like radicals and behave like variables in addition/subtraction.

Conclusion

This calculator simplifies any radical to its most reduced form instantly, making manual calculations easier and faster.

References

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