What Are Radicals in Math?

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A radical, or root, is the mathematical opposite of an exponent, in the same sense that addition is the opposite of subtraction. The smallest radical is the square root, represented with the symbol √. The next radical is the cube root, represented by the symbol ³√. The small number in front of the radical is its index number. The index number can be any whole number and it also represents the exponent that could be used to cancel out that radical. For example, raising to the power of 3 would cancel out a cube root.

General Rules for Each Radical

The result of a radical operation is positive if the number under the radical is positive. The result is negative if the number under the radical is negative and the index number is odd. A negative number under the radical with an even index number produces an irrational number. Remember that though it isn't shown, the index number of a square root is 2.

Product and Quotient Rules

To multiply or divide two radicals, the radicals must have the same index number. The product rule dictates that the multiplication of two radicals simply multiplies the values within and places the answer within the same type of radical, simplifying if possible. For example,

\sqrt[3]{2}× \sqrt[3]{4} = \sqrt[3]{8}

which can be simplified to 2. This rule can also work in reverse, splitting a larger radical into two smaller radical multiples.

The quotient rule states that one radical divided by another is the same as dividing the numbers and placing them under the same radical symbol. For example,

\frac{\sqrt{4}}{\sqrt{8}} = \sqrt{\frac{4}{8}} = \sqrt{\frac{1}{2}}

Just like the product rule, you can also reverse the quotient rule to split a fraction under a radical into two individual radicals.


  • Here's an important tip for simplifying square roots and other even roots: When the index number is even, the numbers inside the radicals can't be negative. In any situation, the denominator of the fraction can't equal out to 0.

Simplifying Square Roots and Other Radicals

Some radicals solve easily as the number inside solves to a whole number, such as √16 = 4. But most won't simplify as cleanly. The product rule can be used in reverse to simplify trickier radicals. For example, √27 also equals √9 × √3. Since √9 = 3, this problem can be simplified to 3√3. This can be done even when a variable is under the radical, though the variable has to remain under the radical.

Rational fractions can be solved similarly using the quotient rule. For example,

\sqrt{\frac{5}{49}} = \frac{\sqrt{5}}{\sqrt{49}}

Since √49 = 7, the fraction can be simplified to √5 ÷ 7.

Exponents, Radicals and Simplifying Square Roots

Radicals can be eliminated from equations using the exponent version of the index number. For example, in the equation √​x​ = 4, the radical is canceled out by raising both sides to the second power:

(\sqrt{x})^2 = (4)^2\text{ or } x = 16

The inverse exponent of the index number is equivalent to the radical itself. For example, √9 is the same as 91/2. Writing the radical in this manner may come in handy when working with an equation that has a large number of exponents.

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