Working with Roots and Radicals

The root of an expression is the reverse of raising it to a power:

**power**is equal to that expression multiplied by itself 2 times.

**root**of that power:

**radicand.**)

**Anatomy of a Radical Expression**

*n*,

*x*must be positive or zero, otherwise the result is not a real number

Radicals

The most common roots to work with are square roots. If no index number is present, the symbol stands for a square root.

However, not every radical is a square root. If there is an index number present other than the number 2, then the root is not a square root.

How Do You Know if the Root is Positive or Negative?

- A positive number times itself will yield a positive number.
- A negative number times itself will yield a positive number.

To show the negative of a square root, a negative sign would have to be placed outside the radical.

Example

To show the negative square root of 25, the negative sign goes outside the radical:

is not a real number

When working with other roots, note that when the index is odd, a negative radicand can be produced by negative factors.

To simplify square roots, we can make use of the

**product rule for square roots**, which states that the product of two square roots is equal to the square root of the product for all real numbers.

**To Simplify the Square Root of a Constant:**

1) Write the constant as a product of the largest perfect square factor and another factor.

2) Use the product rule to write the expression as a product of square roots, with each square root containing one of the factors.

3) Find the square root of the perfect square factor.

**To Simplify Square Roots Containing Variables:**

1) If the expression consists of a variable raised to an even power, the square root of the expression equals the variable raised to one-half of that power.

Examples:

2) If the variable contains an odd power, express it as the product of two factors, one having an exponent 1 and the other a perfect square.

3) Use the product rule to simplify.

**Example:**

Example

Simplify:

Result:

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