Transcript 7.1/7.2 Nth Roots and Rational Exponents

7.1/7.2 Nth Roots and
Rational Exponents
How do you change a power to rational form and vice
versa?
How do you evaluate radicals and powers with rational
exponents?
How do you solve equations involving radicals and
powers with rational exponents?
The Nth root
Radical
Index
Number
n
a a
Radicand
1
n
n>1
The index number becomes the
denominator of the exponent.
Radicals
n
a
• If n is odd – one real root.
• If n is even and
– a>0
– a=0
– a<0
Two real roots
One real root
No real roots
Example: Radical form to
Exponential Form
Change to exponential form.
3
x
2
x
or
2
3
 
 x
1
2
3
or
 x
2

1
3
Example: Exponential to Radical
Form
Change to radical form.
x
2
3
 x
3
2
or
 x
3
2
The denominator of the exponent
becomes the index number of the radical.
Example: Evaluate Without a
Calculator
Evaluate without a calculator.
1.

3

5
8 

3
2
  2 
 32
3
5

5
2.
4
32  4 25
 4 2  24
 24 2
Example: Solving an equation
Solve the equation:
x  7  9993
4
x  7  7  9993  7
4
x  10000
4
4
x  4 10000
4
x  10
Note: index number
is even, therefore,
two answers.
Rules
• Rational exponents and radicals follow the
properties of exponents.
• Also, Product property for radicals
n
a b  n a  n b
• Quotient property for radicals
n
a na
 n
b
b
Example: Using the Quotient
Property
Simplify.
4
16
2
2
4
4
 4 
81
3
3
Adding and Subtracting Radicals
Two radicals are like radicals, if they have the same
index number and radicand
Example
3
3
2 and 4 2 are like radicals.
Addition and subtraction is done with like radicals.
Example: Addition with like radicals
Simplify.
 x 2 x  x
4
4
4
Note: same index number and same radicand.
Add the coefficients.
Example: Subtraction
Simplify.
4x  x x
5
3
Note: The radicands are not the same. Check to
see if we can change one or both to the same
radicand.
4x  x  x x
2
3
3
2x x  x x
3
3
Note: The radicands are the same. Subtract
coefficients.
x x
3