Simplifying Radicals

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Simplifying Radicals Notes

Often when we have a radical expression, we need to simplify it. A radical is in simplified form if it meets 3 criteria: * There are no perfect nth-factors inside the radical * There are no fractions inside a radical * There are no radical signs in the denominator of a fraction
In this section, we will deal only with the first criterion – that there be no perfect nth factors inside the radical sign.

To simplify we will be using the definition of fractional powers:
Definition of Fractional Exponents

We have worked with the first part of this definition before, but we will use the last part in simplifying. Study the following examples:

Example. Use the definition of fractional exponents to rewrite the radical expression in exponential form. 1.
By our definition, we can rewrite this as:

2.
When we have a square root, the index is understood to be 2. We can write in the index and then proceed as above:

Sometimes we can simplify our result after writing in exponential form: 3.
Applying the definition we get . If we simplify the exponent, , and so the problem simplifies to

4.
Applying the definition and simplifying we get

5.
Again, the index of the square root is 2 and so

In addition to the definition of fractional exponents, we will also use the following:

Rule of Radicals 1:

This rule can either be used from left to right or from right to left.

Simplifying Square Root Expressions

In order to simplify a square root, we need to make sure that there are no perfect square factors inside the radical sign. If there are, then we need to simplify. To help us, it is useful to have a list of the perfect squares.

Perfect Squares from 1 to 100:
1 4 9 16 25 36 49 64 81 100

Examples: Simplify the following radicals: 1.
To simplify, we first look…...

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