Radicals and Rational Exponents
Definition of the Principal Square
Root• If a is a nonnegative real number,
the
nonnegative number b such that b2 = a,
denoted by is the principal square
root of a. |
Square Roots of Perfect Squares
![](./articles_imgs/4517/radica51.gif) |
The Product Rule for Square Roots
• If a and b represent nonnegative real
number, then
![](./articles_imgs/4517/radica52.gif)
• The square root of a product is the product
of the square roots. |
Text Example
• Simplify ![](./articles_imgs/4517/radica53.gif)
Solution:
![](./articles_imgs/4517/radica54.jpg) |
The Quotient Rule for Square Roots
• If a and b represent nonnegative real
numbers and b does not equal 0, then
and
![](./articles_imgs/4517/radica56.gif)
• The square root of the quotient is the
quotient of the square roots. |
Text Example •
Simplify:![](./articles_imgs/4517/radica57.gif)
Solution:![](./articles_imgs/4517/radica58.gif) |
Example
• Perform the indicated operation:
![](./articles_imgs/4517/radica59.gif)
Solution:![](./articles_imgs/4517/radica60.gif) |
Example
• Perform the indicated operation:
![](./articles_imgs/4517/radica61.gif)
Solution:
![](./articles_imgs/4517/radica62.gif)
|
Definition of the Principal nth Root
of a Real Number
means that
![](./articles_imgs/4517/radica64.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
nth PowersIf n is odd,
![](./articles_imgs/4517/radica65.gif)
If n is even an ![](./articles_imgs/4517/radica66.gif) |
The Product and Quotient Rules
for nth Roots
• For all real numbers, where the indicated
roots represent real numbers,
and
![](./articles_imgs/4517/radica68.gif) |
Definition of Rational Exponents
![](./articles_imgs/4517/radica69.gif)
Furthermore,
![](./articles_imgs/4517/radica70.gif)
|
Example • Simplify
![](./articles_imgs/4517/radica71.gif)
Solution:
![](./articles_imgs/4517/radica72.gif)
|
Definition of Rational Exponents
![](./articles_imgs/4517/radica73.gif)
• The exponent m/n consists of two parts: the
denominator n is the root and the numerator
m is the exponent. Furthermore,
![](./articles_imgs/4517/radica74.gif)
|
Radicals and
Rational Exponents |
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