Download 7.2 - Rational Exponents

DeSmet - Math 152
Blitzer 5E
∫ 7.2 - Rational Exponents
*Note: From here on out, we will assume ALL variables are positive, i.e., no more absolute values! You will
have slightly different answers in your HW from the book.
1. Remember your exponent rules?
(x )
m n
xm xn =
( xy )m =
=
m
⎛ x⎞
⎜⎝ y ⎟⎠ =
x0 =
xm
=
xn
⎛ x⎞
⎜⎝ y ⎟⎠
x−m =
−n
1
x−m
2. We have used these rules for m and n integers. These rules NEED to hold for rational (fractional)
numbers as well!
3. Rational Exponents
What would it mean to raise 3 to the 1/2?
31/2 ⋅ 31/2 =
51/ 3 ⋅ 51/ 3 ⋅ 51/ 3 =
21/8 ⋅ 21/8 ⋅ 21/8 ⋅ 21/8 ⋅ 21/8 ⋅ 21/8 ⋅ 21/8 ⋅ 21/8 =
1
n
1
n
1
n
a ⋅ a ⋅⋅⋅ a =
n − times
1
n
Thus for x a real number and n a natural number greater than 1, x can be written:
Section 7.2!
pg. 1
DeSmet - Math 152
Blitzer 5E
Example 1:
Write each expression in radical from and simplify the result if possible.
1
( a ) ( 5st )
1 4
(b ) ⎛⎜⎝ ⎞⎟⎠
16
1
8
Example 2: Write
10
2xy 5
z2
and
(
5
6x 2 y
1
( c ) − 64 3
)
8
1
( d ) ( −64 ) 3
in rational exponent form.
Example 3: Simplify each of the following using exponent rules:
1
( a ) ( 27x 3 ) 3
(b )
1
4 2
(100x )
(c)
1
4 4
( −16x )
4. Rational exponents with numerators other than 1:
“Creatively break-up” 16 3/2 and x m /n to investigate the meaning of a rational exponent meaning of the
form
m
:
n
Discovered result:
Section 7.2!
pg. 2
DeSmet - Math 152
Blitzer 5E
Example 4: Simplify each expression:
(i )
36
1
(ii ) ⎛⎜⎝ − ⎞⎟⎠
8
3
2
4/3
(iii ) ( −27x 6 )
2/3
5. Negative Exponents: Again, these are not threat to us, as we the same rules apply!
That is:
x
− m /n
=
1
x − m /n
⎛ x⎞
⎜⎝ y ⎟⎠
=
− m /n
=
Example 5: Simplify each expression (i.e., your final answer must not have negative exponents)
(i )
25
−
3
2
(ii ) ( −64x 6 )
−2 / 3
6. Simplifying Expressions with Rational Exponents:
Again, nothing new because we can use our exponent rules!
Example 6: Simplify each expression, write answers without negative exponents.
1
(i )
4 2/5 4 2/5
(ii )
(iii )
51/35 −4/3
3
7878
5
78
(iv )
(k )
Section 7.2!
3/8 1/8
(v)
(x
)
−3/5 1/4 1/3
y
( 2y )
( vi )
y
1/5 4
3/10
pg. 3
DeSmet - Math 152
7. Using Rational Exponents to Simplify Expressions:
Life (Algebra at least) is easier in rational exponent land. Why?
Blitzer 5E
Thus, we can convert into rational exponents land, do work, and return to radical land.
Example 7: Use rational exponents to simplify each expression. If rational exponents appear after
simplifying, write the answer in radical notation. Assume all variables represent positive numbers.
(i )
(ii )
q2
8
49x 2 y 2
4
(iii )
9
(iv )
4 3
(v)
x6 y3
2x
y2
6 y
3
8. Application Problems have involve plugging in numbers into a radical function and giving an
approximate solution using your calculator. Can you with your calculator!? I.e. How could you
approximate 12 ⋅ 51/3 ?
Section 7.2!
pg. 4