2.6 Radical and Rational Exponent Equations

Radical Equations Notes

How to find a 3rd, 4th, 5th, etc. root on a calculator

We'll find the 4th root of 81

• Figure out if your calculator has ^ or y^x as the exponent button

• Select the following buttons if your calculator has ^

• 4 > 2nd > ^ > 81 > =

• Select the following buttons if your calculator has y^x

• 81 > 2nd > y^x > 4 > =

• You should get 3 as your answer

Multiple Choice

How do you think you would get x by itself in

$$\sqrt{x}=3$$  ?

• square both sides

• take the square root of both sides

• multiply by the square root

• multiply by 2

Multiple Select

What did I say are the two big steps in these radical equation problems?

• get rid of the fractions

• isolate the radical

• move everything to one side of the equal sign

• get rid of the radical

Multiple Choice

What should you do first to try to solve this equation? $^3\sqrt{2x-5}-6=-3$  

• subtract -5-6

• add 6 to the other side

• subtract 6 to the other side

• divide by -6

Multiple Choice

Now what? 

 $^3\sqrt{2x-5}=3$  

• square both sides

• cube both sides

• divide by 3

• add 5

• take the cube root of both sides

Multiple Choice

What is the result when you cube both sides?

 $^3\sqrt{2x-5}=3$  

•  $8x^3-125=27$  

•  $2x^3-125=27$  

•  $2x-5=3$  

•  $2x-5=27$  

Multiple Choice

Solve for x.

 $2x-5=27$  

• 64

• 11

• 16

• 6

Multiple Choice

When can extraneous solutions arise?

• in any kind of radical equation

• when you raise both sides to an even exponent

• when you raise both sides to an odd exponent

• when you do the problem wrong

Multiple Choice

True or False:

(x+6)² = x²+36

• True

• False

Multiple Choice

What should you do first to solve for x?

 $1+^4\sqrt{x-2}=3$  

• add 1

• square both sides

• raise both sides to the 4th power

• subtract 1

• add 2

Multiple Choice

Now what?

 $^4\sqrt{x-2}=2$  

• raise both sides to the 4th power

• add 2 to the other side

• divide by 2

• divide by 4

• square both sides

Multiple Choice

Solve for x.

 $x-2=2^4$  

• 6

• 10

• 16

• 18

• 8

Multiple Choice

Check your answer x=18 by plugging it into the original equation. $$1+^4\sqrt{x-2}=3$$  

• it's valid

• it's extraneous

Multiple Choice

 $x-3=\sqrt{-2x+41}$  

Solve for x.

• 8

• -4

• 4

• no solution

• -8

Multiple Choice

$$x-3=\sqrt{-2x+41}$$   What is the extraneous solution?

• 8

• 4

• -4

• -8

• none

Terms to know

• Radical Symbol: The line in the middle

• Index: This tells you the root you are taking. The number of times you need the radicand to repeat

• Radicand: everything inside the radical symbol

• Exponent: Tells you how many times something is repeated and multiplied

Terms to know

• We can always rewrite a radical expression into powers using fractions as exponents (aka rational exponents)

• You will need the radicand to act as your base, the index to act as your denominator and the exponent to act as your numerator

Multiple Choice

Select the equivalent form of $$^3\sqrt{x^8}$$  

• $$x^{\frac{8}{3}}$$  

• $$x^{\frac{3}{8}}$$  

To cancel out an exponent that's a fraction, raise it to the reciprocal of the exponent.

Multiple Choice

What power would work to cancel the exponent?

( x^2/3 )^? = x¹

• 2/3

• -2/3

• -3/2

• 3/2

Multiple Choice

Solve the equation $$x^{\frac{7}{3}\ }=\ 128$$   

• 8
  

• 16

• 512

• 82570

Rational Exponent Equations

• Isolate the radical/exponential term FIRST

• THEN, use reciprocal powers to solve

Multiple Choice

$$1024\ =\left(15n\ +\ 4\right)^{\frac{5}{3}}\$$  

• $$4$$  

• $$6935.2$$  

• $$-48\ and\ \ 80$$  

• $$\pm\ 4$$