Tips for Multiplying Radicals

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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}