FST 6-1/6-2 Notes

FST 6-1/6-2 Notes

Convert & simplify between radicals & rational exponents, find domain & range of radical functions, use exponent properties to evaluate and simplify both radicals and rational exponents, find domain and range of/use power functions

Rules for radicals

•  $\left(^n\sqrt{a}\right)^n=a$  

•  $^n\sqrt{a\cdot b}=^n\sqrt{a}\cdot^n\sqrt{b}$  

•  $^n\sqrt{\frac{a}{b}}=\frac{^n\sqrt{a}}{^n\sqrt{b}}$  

•  $^n\sqrt{a^p}=\left(^n\sqrt{a}\right)^p=a^{\frac{p}{n}}$  

•  $^m\sqrt{^n\sqrt{a}}=^{mn}\sqrt{a}=a^{\frac{1}{m\cdot n}}$  

• Radicand cannot be negative if n is even (imaginary #) 

• Radicand can be negative if n is odd (real #)

Multiple Select

Solve: x⁴ = 4096. Pick 2 answers.

• 8

• 4

• -4

• -8

• 16

Multiple Choice

Evaluate

 $1000^{\frac{1}{3}}$  

• -10

• 100

• 10

• 5

Multiple Choice

Evaluate

 $^3\sqrt{-216}$  
 $Hint:\ 6^3=216,\ 7^3=343,\ 8^3=512,\ 9^3=729$  

• -6

• -7

• -8

• 6

• 7

Multiple Choice

For 0 < x < 1, which statement is true?

• $x^{\frac{1}{2}}=x^{\frac{1}{7}}$

• $x^{\frac{1}{2}}<x^{\frac{1}{7}}$

• $x^{\frac{1}{2}}>x^{\frac{1}{7}}$

Multiple Choice

For x > 1, which statement is true?

•  $x^{\frac{1}{3}}=x^{\frac{1}{7}}$ 

•  $x^{\frac{1}{3}}<x^{\frac{1}{7}}$ 

•  $x^{\frac{1}{3}}>x^{\frac{1}{7}}$ 

Multiple Choice

Find the inverse of

 $f\left(x\right)=\frac{1}{4}x-7$  
Hint) 

Step 1: Switch x & y.
Step 2: Isolate y.

• f^-1(x) = 4x + 7

• f^-1(x) = -4x + 28

• f^-1(x) = -4x - 7

• f^-1(x) = 4x + 28

Cone, sphere, cube

• SA for a cone:

   $SA=\pi\cdot r\cdot\sqrt{r^2+h^2}$  

• Volume of a cone:  $V\ =\ \frac{1}{3}\pi\cdot r^2\cdot h$  

• Volume of sphere:  $V=\frac{4}{3}\pi\cdot r^3$  

•  $Volume\ of\ a\ cube:\ V\ =\ x^3$  

•  $Surface\ area\ of\ a\ cube:\ SA=6x^2$  

Cone, sphere, cube

• Volume of a cone:  $V\ =\ \frac{1}{3}\pi\cdot r^2\cdot h$ 

• Ex) Give a formula for the radius in terms of V if height = radius.  $V\ =\ \frac{1}{3}\pi\cdot r^2\cdot h=\frac{\pi r^3}{3}\ due\ to\ h\ =\ r.$  

•  $Step\ 1:\ Multiply\ by\ \frac{3}{\pi}\ to\ both\ side;\ \frac{3V}{\pi}=r^3$   $Step\ 2:\ Isolate\ r\ by\ taking\ ^3\sqrt{ }\ to\ both\ sides;\ ^3\sqrt{\frac{3V}{\pi}}=r$  

Multiple Choice

Given f(x) = 3x + 10

and g(x) = x - 2

Find f(g(5))

• 19

• 23

• -10

• 25

• None of these

Multiple Choice

f(x) = 3x + 10

g(x) = x - 2

Find g(f(-10))

• -42

• -26

• -18

• -22

• None of these

Multiple Choice

Given f(x) = 2x and

g(x) = x²+3 , find f(g(x)).

• x²+2x+3

• 4x²+3

• 2x²+3

• 2x²+6

Poll

Did you find this lesson interesting & challenging?

• A) Too hard

• B) Just right

• C) Too easy

Open Ended

Write a summary or write the formula for the volume/surface of a cube or cone.