HOME
adding fractions
brackets
comparing fractions
complex fractions
decimals
difference squares
dividing fractions
equations
equivalent fractions
exponents
factoring
factoring polynomials
factoring trinomials
finding least common multiples
fractions
fractions decimals
graphing linear equations
inequalities
linear equations
linear inequalities
multiplying dividing fractions
multiplying fractions
multiplying polynomials
percents
polynomials
powers
powers roots
quadratic equations
quadratic expressions
radicals
rational expressions
rational inequalities
rationalizing denominators
reducing fractions
roots
roots radicals
simplifying complex fractions
simplifying fractions
solving equations
solving linear equations
solving quadratic equations
solving radical equations
solving systems linear equations
straight lines
subtracting fractions
systems linear equations
trinomial squares

Roots

Roots

For even values of n , the expression is defined to be the positive nth root of a or the principal nth root of a. For example, denotes the positive second root, or square root, of a , while is the positive fourth root of a. When n is odd, there is only one n th root, which has the same sign as a. For example, , the cube root of a, has the same sign as a. By definition, if then . On a calculator, a number is raised to a power using a key labeled For example, to take the fourth root of 6 on a TI-83 calculator, enter , to get the result 1.56508458.

EXAMPLE 1

Calculations with Exponents

Rational Exponents

In the following definition, the domain of an exponent is extended to include all rational numbers.

DEFINITION OF

For all real numbers a for which the indicated roots exist, and for any rational number m/n

EXAMPLE 2

Calculations with Exponents

NOTE could also be evaluated as but this is more difficult to perform without a calculator because it involves squaring 27, and then taking thecube root of this large number. On the other hand, when we evaluate it as we know that the cube root of 27 is 3 without using a calculator, and squaring 3 is easy.

All the properties for integer exponents given in this section also apply to any rational exponent on a nonnegative real-number base.

EXAMPLE 3

Simplifying Exponential Expressions

In calculus, it is often necessary to factor expressions involving fractional exponents.

EXAMPLE 4

Simplifying Exponential Expressions

Factor out the smallest power of the variable, assuming all variables represent positive real numbers.

Solution

To check this result, multiply by

Solution

The smallest exponent here is 3. Since 3 is a common numerical factor, factor out

Check by multiplying. The factored form can be written without negative exponents as

.

Solution

There is a common factor of 2. Also, have a common factor. Always factor out the quantity to the smallest exponent. Here -1/2 < 1/2 so the common factor is and the factored form is

Buy  Algebrator now: 

Instant download and optional CD

Only $39.99

Click to Buy Now:



OR

2Checkout.com is an authorized reseller
of goods provided by Softmath

Attention: We are currently running a special promotional offer for algebra-helper.com Visitors -- if you order Algebrator by midnight of March 10th you will pay only $39.99 instead of our regular price of $74.99 -- this is $35.00 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the left, not through our regular order page.

If you order now you will also receive 30 minutes of live math tutoring from tutor.com!

You Will Learn Algebra Better - Guaranteed!

Just take a look how incredibly simple Algebrator is:

Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:

Step 2 : Let Algebrator solve it:

Step 3 : Ask for an explanation for the steps you don't understand:

Algebrator can solve problems in all the following areas:

  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • operations with complex numbers (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)

Buy  Algebrator now: 

Instant download and optional CD

Only $39.99

Click to Buy Now:



OR

2Checkout.com is an authorized reseller
of goods provided by Softmath

2007-11-20 08:49:29