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### Rules of Exponents

```
Date: 10/02/1999 at 18:24:22
From: CHRIS HELTON
Subject: Rules of indices

The question I am having trouble with is:

Use the rules of indices to simplify the following expression:

(16x^2)^(1/4)*(4x^3)^(1/2)
--------------------------
(27^4)^(1/3)

I don't know where to start breaking the question down to attempt to
simplify it. This question is part of the foundation topic assignment
given to me in the class titled Mathematics 1 at the University of
Strathclyde.
```

```
Date: 10/11/1999 at 21:29:59
From: Doctor Sandi
Subject: Re: Rules of indices

Hi Chris,

First of all I want to go through some of the log laws with you.

Consider 64 = 4^3

The 3 is called the exponent; the 4 is called the base. This can also
be expressed as log(base 4) 64 = 3.

Something else worth committing to your permanent memory is that any
number raised to the power of 1/2 is the same as the square root of
that number. For example, 4^(1/2) = sqrt(4) = 2. This also happens
with numbers raised to the power of 1/3: the answer will be the same
as the cube root of the number.

Here are some rules for dealing with indices (write these out properly
without the ^ so that they don't look so clumsy):

1.  a^m * a^n = a^(m+n)
2.  a^m / a^n = a^(m-n)
3.  (a^m)^n   = a^(mn)
4.  (ab)^m    = (a^m)(b^m)
5.  a^0       = 1
6.  a^(-n)    = 1/(a^n), a is not equal to 0
7.  a^(1/q)   = q-root(a)
8.  a^(p/q)   = q-root (a^p) = (q-root(a))^p

Generally when simplifying index expressions:

1. Express all terms in index form (lowest base) where necessary
2. Remove the brackets
4. Express with positive indices

When you have an index raised to the power of another index, you
multiply the indices. You end up with:

First the numerator (top half of the fraction)

[(16^(1/4))(x^(2*(1/4))][(4^(1/2)(x^(3*1/2)]
= [2x^(1/2)][2x^(3/2)]
= (4x^[(1/2)+(3/2)]
= (4x^2)

then the denominator:

(27^4)(^1/3)

Let's deal with the power first. It is a power raised to another power
so we will multiply them together and end up with:

4 * (1/3) = 4/3

So we have

27^(4/3) = 81

(4x^2)/81

I hope that this has been helpful for you. If you have any other other
questions that you would like to ask Dr. Math, please feel free to
write back.

- Doctor Sandi, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Exponents

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