Radicals and roots explained
A radical is a root, the inverse of a power. This guide explains what radicals mean, how to simplify square roots, the link between radicals and fractional exponents, and how to rationalize a denominator.
What a radical is
The square root of 9 is 3 because 3 squared is 9. The radical sign asks what number, raised to the given power, gives the value inside. Square roots are the most common, but cube roots and higher roots follow exactly the same idea.
Simplifying square roots
- Find the largest perfect-square factor of the number under the root.
- Split the radical into that perfect square times the rest.
- Take the square root of the perfect square and move it outside.
Simplify √72. Since 72 = 36 × 2, √72 = √36 × √2 = 6√2.
Radicals and fractional exponents
Every radical can be written as a fractional exponent. The square root of a is a to the one-half; the cube root of a is a to the one-third; the nth root of a is a to the one over n. This link lets you apply the exponent rules to roots, which is often the easiest way to simplify complicated expressions.
Rationalizing the denominator
By convention, a radical is not left in the denominator of a fraction. To remove it, multiply the top and bottom by that radical. So 1 over root 2 becomes root 2 over 2 after multiplying both parts by root 2. This rationalizing step gives a tidier, standard form.
- A radical is a root, the inverse operation of a power.
- Simplify a square root by pulling out the largest perfect-square factor.
- The nth root of a equals a to the power 1/n.
- Exponent rules apply to radicals through fractional exponents.
- Rationalize by clearing radicals from the denominator.
Related guides
See the Exponents Guide and the Squares and Square Roots Table.
FAQ
What is a radical in math?
A root, such as a square root or cube root; the inverse of raising a number to a power.
How do I simplify a square root?
Factor out the largest perfect square. For √72, that is 36, giving 6√2.
How are radicals related to exponents?
A radical is a fractional exponent: the nth root of a is a^(1/n).