## Algebra

Showing posts with label index. Show all posts
Showing posts with label index. Show all posts

### Rational Exponents

Rational (fractional) exponents, indicate that we have a radical expression.

They are intimidating at first but you will quickly get used to working with them. The denominator is the index and the numerator is the exponent.

[ Video Instruction: Rational Exponents ]

Rewrite the radical expression using rational exponents.

Convert to a radical and then simplify.

When simplifying, you can use the rules of exponents to simplify or you can convert to a radical and then simplify. Both methods yield the same result.

Assume all variables are positive and use the rules of exponents to simplify.
Simplify. (Assume all variables are positive.)

Notice that all of the above problems worked out nicely because the exponents were multiples of the root.  This is not always the case as illustrated below.

A quick way to simplify radicals is to divide the index into the exponents. This will tell us what the exponent of the base should be outside the radical and the remainder will be the exponent of the base inside the radical. All the rules of exponents hold true when working with rational exponents.

Sometimes we will be asked to multiply radicals with different indices.  This may seem impossible at first, but it can be done using the rules of exponents as follows.

### Multiplying and Dividing Radical Expressions

As long as the indices are the same, we can multiply the radicands together using the following property.

Since multiplication is commutative, you can multiply the coefficients and the radicands together and then simplify.

Multiply.

Take care to be sure that the indices are the same before multiplying.  We will assume that all variables are positive.

Simplify.

Divide radicals using the following property.
Divide. (Assume all variables are positive.)
Rationalizing the Denominator
A simplified radical expression cannot have a radical in the denominator.  When the denominator has a radical in it, we must multiply the entire expression by some form of 1 to eliminate it.  The basic steps follow.

Rationalize the denominator:
Multiply numerator and denominator by the 5th root of of factors that will result in 5th powers of each factor in the radicand of the denominator.
Notice that all the factors in the radicand of the denominator have powers that match the index.  This was the desired result, now simplify.
Rationalize the denominator.
This technique does not work when dividing by a binomial containing a radical.  A new technique is introduced to deal with this situation.

Rationalize the denominator:
Multiply numerator and denominator by the conjugate of the denominator.
And then simplify. The goal is to eliminate all radicals from the denominator.

Rationalize the denominator.