5.1 Properties of Exponents – Day 1 (6.1, 7.6)
Objective: TSW simplify properties of exponents with integer exponents
Simplifying completely:
1. There are no powers of powers
2. Each base appears only once
3. All fractions are in simplest form.
4. There are no negative exponents
Example: Simplify
a)
b)
c)
You Try:
a)
b)
c)
d)
Example: Simplify
a)
b)
5.1 Properties of Exponents – Day 2 (6.1, 7.6)
Objective: TSW simplify properties of exponents with rational exponents
It is just like yesterday but….you have fractions for exponents!!!!
Remember: 1)
2)
3)
4)
There are no powers of powers
Each base appears only once
All fractions are in simplest form
There are no negative exponents
Example: Simplify
a)
1
4 ⁄2
∙
5
4 ⁄2
b)
4
16 ⁄3
1
16 ⁄3
c)
𝑥
1⁄
2
∙𝑥
4⁄
3
d) 𝑦 −
2⁄
3
∙𝑦
−1⁄
3
Example: Simplify
a)
2
𝑥 ⁄3 𝑥 −1
1
𝑥 ⁄3
−1⁄
4
5⁄
1⁄
b) 𝑥 2 (𝑥 2 ) 2
c)
(𝑥 4 𝑦 2 )
2
(𝑥 −2 𝑦 3 ) ⁄3
5.2 Convert Rationals and Radicals (7.4, 7.5)
Objective: TSW evaluate and simplify rational exponents. TSW convert rational exponents to radicals
Radical Form of am/n
If all indicated roots are real numbers, then
That is, raise a to the mth power and then take the nth root,
or
take the nth root of a and then raise to the mth power.
EXAMPLE 1 Evaluate each exponential.
a. 321/ 5
b. 641/ 2
c.
1/ 3
d.  1 
(81)1/ 4
 
 27 
EXAMPLE 2 Evaluate each exponential.
a.
253/ 2
b. 27 2 / 3
c. 163/ 2
d.  64 
2/3
e.
EXAMPLE 3 Write all radicals as exponentials, and then apply the rules for rational exponents. Leave answers in
exponential form.
a.
x5
3
x
b.
Recall: Simplifying Radicals
Example: Simplify
a)
√18
d)
√25𝑝7
3 6
x
3
b) √32
e)
3
c) √54
27 y 7 x5 z 6
f)
 4 32a5b7
Now let’s apply 5.1 and 5.2 information:
Example: Simplify – Write your answer in simplest root and exponential form.
1⁄
2
a) (𝑥 3 𝑦 7 𝑧 8 )
5
1⁄
3 (2𝑥 ⁄3 )
b) (8𝑥 5 )
2⁄
3
c) (−125𝑥 4 √𝑥)
5.2 Solve Radical Equations – Day 1 (7.7)
Objective: TSW solve radical equations including square roots and cubed roots
Solving an Equation with Radicals
Step 1 Isolate the radical. Make sure that one radical term is alone on one side of the equation.
Step 2 Apply the power rule. Raise both sides of the equation to a power that is the same as the index of the
radical.
Step 3 Solve the resulting equation; if it still contains a radical, repeat Steps 1 and 2.
Step 4 Check all proposed solutions in the original equation.
Example: Solve
a) √𝑥 − 2 + 5 = 8
b) 3√2𝑥 − 1 − 9 = −3
Example: Solve
3
a) √𝑥 − 5 = 2
b) √5𝑥 − 8 − 1 = 0
3
c) √𝑥 + 1 + 2 = −4
5.3 Solve Radical Equations – Day 2 (7.7)
Objective: TSW solve radical equations with multiple roots
Solving an Equation with Radicals
Step 1 Isolate the radical. Make sure that one radical term is alone on one side of the equation.
Step 2 Apply the power rule. Raise both sides of the equation to a power that is the same as the index of the
radical.
Step 3 Solve the resulting equation; if it still contains a radical, repeat Steps 1 and 2.
Step 4 Check all proposed solutions in the original equation.
Example: Solve
a) √𝑥 − 4 = √2𝑥 − 13
b)
c) 𝑥 = √𝑥 2 − 5𝑥 + 15
d)
e)
1  2 p  p 2  p  1.
f)
3
2 x  7  3 3x  2.
5  x  x  1.
2 x  3  x  1  1.
5.4 Solve Rational Equations – Day 1 (7.7)
Objective: TSW solve rational equations where the numerator is one.
Rational Equation: fractional exponents....
Remember a fractional exponent just represents a root!
Example: Solve
1
a)
1
(3𝑥 − 1)2 = (2𝑥 + 5)2
1
d) (3𝑥 + 1)3 = −5
1
b) 3(5𝑥 − 1)2 − 2 = 0
1
e) 3(2𝑥 + 6)4 − 6 = 0
1
c) (3𝑥 + 2)3 + 1 = 0
5.4 Solve Rational Equations – Day 2 (7.7)
Objective: TSW solve rational equations where the numerator is an integer
Rational Equation: fractional exponents....
Remember a fractional exponent just represents a root!
Today's Rational equations are going to be in the above form
Example: Solve
a) (𝑥
c)
2
3
− 1)2 = 64
3
4
3(𝑥 + 2) + 6 = 30
B)
4(3𝑥 + 5)3 = 100
Example: A formula used to determine the velocity v, in feet per second, 𝑣 = √2𝑔𝑑,
of an object after it has fallen a certain distance is where g is the acceleration due to gravity and d is the distance the
object has fallen. On Earth, the acceleration (g) due to gravity is approximately 32 feet per second.
a) Find the velocity of a person after falling 3 feet.
b)
How many feet has the object fallen when it is traveling at 90 feet per second?
c)
If the object was dropped from the top of a building that is 600 feet above ground. How fast was it traveling at
the time that it hit the ground?
5.5 Find Inverses of any Polynomial (7.2)
Objective: TSW find the inverse of a function and graph. TSW determine if two functions are inverses of each other.
If two functions are inverses, they reflect over the identity function, y = x.
EXAMPLE:
x
EXAMPLE:
x
EXAMPLE: