**Adding and Subtracting Radicals**

Two radicals can be added or subtracted if they have the same radicand. To add two radicals with the same radicand, add the coefficients of the radicals and keep the radicand the same.

2√2 + 3√2 = 5√2

To subtract two radicals with the same radicand, subtract the coefficient of the second radical from the coefficient of the first radical and keep the radicand the same.

6√5 –4√5 = 2√5

The expressions √3 + √2 and √3 – √2 cannot be simplified any further, because these radicals have different radicands.

**Multiplying Radicals**

Two radicals can be multiplied whether they have the same radicand or not. To multiply two radicals, multiply the coefficients of the radicals and multiply the radicands.

(4√6)(3√7) =12√42, because (4)(3) =12 and (√6)(√7)=√42.

If two radicals of the same root with the same radicand are multiplied, the product is equal to the value of the radicand alone.

Here's an example: (√6)(√6) = 6. Both radicals represent the same root, the square root, and both radicals have the same radicand, 6, so the product of √6 and √6 is 6.

**Dividing Radicals**

Two radicals can be divided whether they have the same radicand or not. To divide two radicals, divide the coefficients of the radicals and divide the radicands.

= 5√5, because = 5 and = √5.

Any radical divided by itself is equal to 1: = 1.

**Simplifying a Single Radical**

To simplify a radical such as √64, find the square root of 64. Look for a number that, when multiplied by itself, equals 64. Because (8)(8) = 64, the square root of 64 is 8: √64 = 8; √64 is expressed as 8, not –8. The equation *x*^{2} = 64 has two solutions, because both 8 and –8 square to 64, but the square root of a positive number is always its principal root (a positive number) when one exists.

However, most radicals cannot be simplified so easily. Many whole numbers and fractions do not have roots that are also whole numbers or fractions. You can simplify the original radical, but you will still have a radical in your answer.

To simplify a single radical, such as √32, find two factors of the radicand, one of which is a perfect square: √32 = (√16)(√2). Notice that √16 is a perfect square; the positive square root of 16 is 4. So, √32 = (√16)(√2) = 4√2.

**Rationalizing Denominators of Fractions**

An expression is not in simplest form if there is a radical in the denominator of a fraction. For example, the fraction : is not in simplest form. Multiply the top and bottom of the fraction by the radical in the denominator: Multiply by . Because = 1, this will not change the value of the fraction. Because any radical multiplied by itself is equal to the radicand, (√3)(√3) = 3; (4)(√3) = 4√3, so the fraction : in simplest form is :.

**Solving Equations with Radicals**

Use the properties of adding, subtracting, multiplying, dividing, and simplifying radicals to help you solve equations with radicals. To remove a radical symbol from one side of an equation, you can raise both sides of the equation to a power. Remove a square root symbol from an equation by squaring both sides of the equation. Remove a cube root symbol from an equation by cubing both sides of the equation.

If √*x* = 6, what is the value of *x*?

To remove the radical symbol from the left side of the equation, square both sides of the equation. In other words, raise both sides of the equation to the power that is equal to the root of the radical. To remove a square root, or second root, raise both sides of the equation to the second power. To remove a cube root, or third root, raise both sides of the equation to the third power.

√*x* = 6, (√*x*)^{2} = (6)^{2}, *x* = 36

= 3, ()^{3} = (3)^{3}, *x* = 27

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