Radicals and Rational Exponents
Definition of the Principal Square
Root
• If a is nonnegative real number, the
nonnegative number b such that ![](./articles_imgs/4585/radica12.gif)
denoted by
,is the principle square
root of a
Square Roots of Perfect Square
![](./articles_imgs/4585/radica14.gif)
The Product Rule for Square Roots
• If a and b represent nonnegative real
number, then
and
![](./articles_imgs/4585/radica16.gif)
• The square root of product is the product
of the square roots
Text Example
• Simple : ![](./articles_imgs/4585/radica17.jpg)
solution :
![](./articles_imgs/4585/radica18.jpg)
The Quotient Rule for Square Roots
• If a and b does represent nonnegative real
numbers and b does no equal 0, then
and
![](./articles_imgs/4585/radica20.gif)
• The square root of quotient is the
quotient of the square roots
Text Example
• Simple :![](./articles_imgs/4585/radica21.gif)
solution :![](./articles_imgs/4585/radica22.gif)
Example
• Perform the indicated operation:
![](./articles_imgs/4585/radica23.jpg)
solution :
![](./articles_imgs/4585/radica24.jpg)
• Perform the indicated operation:
![](./articles_imgs/4585/radica25.jpg)
solution :
![](./articles_imgs/4585/radica26.jpg)
Definition of the Principal nth Root
of a Real Number
means that
![](./articles_imgs/4585/radica28.gif)
• If n ,the index , is even , then a is
nonnegative (a≥0)and b is also
nonnegative (b≥0).If n is odd ,a and b
can be any real numbers.
Finding the nth Roots of Perfect
th Powers
If n is odd ,![](./articles_imgs/4585/radica29.gif)
If n is even ![](./articles_imgs/4585/radica30.gif)
The Product and Quotient Rules
for nth Roots
• For all real numbers ,where the indicated
roots represent real numbers,
and
![](./articles_imgs/4585/radica32.jpg)
Definition of Rational Exponents
![](./articles_imgs/4585/radica33.gif)
Furthermore
![](./articles_imgs/4585/radica34.jpg)
Example
• Simple :![](./articles_imgs/4585/radica35.gif)
solution:
![](./articles_imgs/4585/radica36.gif)
Definition of Rational Exponents
![](./articles_imgs/4585/radica37.jpg)
•The exponents m/n consists of two parts :the
denominator n is the root and the numerator
m is the exponent. Furthermore .
![](./articles_imgs/4585/radica38.gif)