Tips for Multiplying Radicals

••• RUBEN RAMOS/iStock/GettyImages

A radical is basically a fractional exponent and is denoted by the radical sign (√). The expression ​x2 means to multiply ​x​ by itself (​x​ × ​x​), but when you see the expression √​x​, you're looking for a number that, when multiplied by itself, equals ​x​. Similarly, 3√​x​ means a number that, when multiplied by itself ​twice,​ equals ​x​, and so on. Just as you can multiply numbers with the same exponent, you can do the same with radicals, as long as the superscripts in front of the radical signs are the same. For example, you can multiply (√​x​ × √​x​) to get √(​x2), which just equals ​x​, and (3√​x​ × 3√​x​) to get 3√(​x2). However, the expression (√​x​ × 3√​x​) can't be simplified any further.

Tip #1: Remember the "Product Raised to a Power Rule"

When multiplying exponents, the following is true:

(a)^x × (b)^x = (a × b)^x

The same rule applies when multiplying radicals. To see why, remember that you can express a radical as a fractional exponent. For example,

\sqrt{a} = a^{1/2}

or, in general,

\sqrt[x]{a} = a^{1/x}

When multiplying two numbers with fractional exponents, you can treat them the same as numbers with integral exponents, provided the exponents are the same. In general:

\sqrt[x]{a} × \sqrt[x]{b}= \sqrt[x]{a × b}

Example:​ Multiply √25 × √400

\sqrt{ 25} × \sqrt{400} = \sqrt{25 × 400} = \sqrt{10,000}

Tip #2: Simplify the Radicals before Multiplying Them

In the above example, you can quickly see that

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

and that

\sqrt{400} = \sqrt{20^2}=20

and that the expression simplifies to 100. That's the same answer you get when you look up the square root of 10,000.

In many cases, such as in the above example, it's easier to simplify numbers under the radical signs before you perform the multiplication. If the radical is a square root, you can remove numbers and variables that repeat in pairs from under the radical. If you're multiplying cube roots, you can remove numbers and variables that repeat in units of three. To remove a number from a fourth root sign, the number must repeat four times and so on.

Examples

1.​ Multiply ​√18 × √16

Factor the numbers under the radical signs and put any that occur twice outside the radical.

\sqrt{18} = \sqrt{9 × 2} = \sqrt{3 × 3} × 2 = 3\sqrt{2} \\ \sqrt{16} = \sqrt{4 × 4} = 4 \\ \,\\ \implies \sqrt{18} × \sqrt{16} = 3 \sqrt{2} × 4 = 12 \sqrt{2}

2. Multiply

\sqrt[3]{32x^2 y^4} × \sqrt[3]{50x^3y}

To simplify the cube roots, look for factors inside the radical signs that occur in units of three:

\sqrt[3]{32x^2y^4}= \sqrt[3]{(8 × 4)x^2y^4} = \sqrt[3]{[(2 × 2 × 2) × 4]x^2 (y × y × y)y} = 2y\sqrt[3]{4x^2y} \\ \,\\ \sqrt[3]{50 x^3y} = \sqrt[3]{50 (x × x × x)y} = x\sqrt[3]{50y}

The multiplication becomes

2y\sqrt[3]{4x^2y} × x\sqrt[3]{50y}

Multiplying like terms and applying the Product Raised to Power Rule, you get:

2xy × \sqrt[3]{200x^2y^2}

Related Articles

What Are Radicals in Math?
How to Divide Radicals
Laws of Exponents: Powers & Products
How to Check Multiplication
How to Factor & Simplify Radical Expressions
How to Get Rid of Cubed Power
How to Simplify Monomials
How to Manipulate Roots & Exponents
How to Divide Exponents With Different Bases
How to Factor With Negative Fractional Exponents
Negative Exponents: Rules for Multiplying & Dividing
What Is an Integer in Algebra Math?
How to Simplify Exponents
How to Solve Polynomial Equations
How to Factor Polynomials With 4 Terms
Rules for Multiplying Scientific Notation
Math Rules for Addition
The Basics of Square Roots (Examples & Answers)
How to Simplify Algebraic Expressions
How to Identify a Numerical Coefficient of a Term