Question

Solve the equation 9 = (square root of 27)^(4x + 6). Using complete sentences, explain the strategy used in solving this equation.

Answer 1

Can you first write in the form of equation like using numeral values..?

Answer 2

\[9=\sqrt{27}^{4x+6}\]

Answer 3

Hmm okay, I will not be able to just write the whole strategy for you but I will explain you

Answer 4

okay thanks

Answer 5

Our first and foremost step will be to make each term in the powers of 3 so that we can equate the powers later on.

Answer 6

Can you tell me, what is 9 in the powers of 3?

Answer 7

is it 729?

Answer 8

No, I meant that : 9 = 3^x

What is x?

Answer 9

9 in the powers of 3 means 9 in terms of 3 to the power of something

if it was 9 to the powers of 3 then it would have been 729 but that is not the case for now.

Answer 10

2

Answer 11

Okay good, so : \(\color{blue}{\bf{9 = 3^2}}\)

Answer 12

So, I will write the equation as :

\(\large \color{blue}{\bf{3^2}} = \sf{\sqrt{27}^{4x+6}}\)

Answer 13

okay

Answer 14

so then do you take the square root of 27?

Answer 15

So, now we have to represent the Right Hand Side in the TERMS OF POWERS OF 3 :

\(\large \sqrt{27}^{4x+6} = 3^y \)

So, to calculate what is y, we have to do a lot of hardwork. Ready captain? :P

Answer 16

yea I'm ready

Answer 17

Good! One medal for that to you!

Answer 18

Okay now, let us get back to work.

\(\sqrt{27} \) , it can be written as : \(\sqrt{3 \times 3 \times 3 }\)

Answer 19

So, it becomes : \(\sqrt{27} = \boxed{3\sqrt{3}}\)

Right?

Answer 20

right

Answer 21

Okay so, we can right : \(\sqrt{3} \) as \(3^{\cfrac{1}{2}}\)
Right?

Answer 22

So, it becomes : \(3\sqrt{3} = 3 \times 3^\frac{1}{2} \)

Can you simplify it?

Answer 23

10 1/2

Answer 24

Oh no..

See it is a theorem : \(\large a^n \times a^m = a^{m+n} \)

Answer 25

im confused

Answer 26

Okay, I will help you to get out from confusion.

Answer 27

See, I have : \(\sqrt{27}\) as \(3\sqrt{3}\) = \(3^1 \times 3^\frac{1}{2}\)

Now, as 3 is common I can write it as : \(3^{1 + \frac{1}{2}}\)

What I did is just added up the powers as the base was same..

Answer 28

okay

Answer 29

so it would be \[3^{5/4}\]

Answer 30

So, it is : \(\large{ 3^{1+ \cfrac{1}{2}}}\)

Answer 31

typo sorry i mean 5/2

Answer 32

No Shay. it is neither 5/2 nor 5/4

1 + 1/2 = (2+1)/2 = 3/2

Answer 33

So, it will be : \(\large{3^\cfrac{3}{2}}\)

Did you get it now?

Answer 34

yea i did the adding wrong

Answer 35

Okay, no problem, it happens!

Answer 36

So, what we have now is :

\(\sqrt{27} = 3^{\cfrac{3}{2}}\)

Answer 37

okay

Answer 38

So, plugin this value in the given equation :

9 = \(\sqrt{27} ^{4x+6}\)

Can you tell me what we will have the equation as now?

Answer 39

Just plugin the values of 9 and \(\sqrt{27}\) that we found above in the equation/.

Answer 40

\[3^{2}=3\frac{ 3 }{ 2 }^{4x+6}\]

Answer 41

Okay well it will be : \(3^2 = 3^{\cfrac{3}{2}(4x+6)}\)

Answer 42

So, another theorem in use comes here :

Whenever the base is same , just equate the powers like : \(a^m = a^n\) then m = n

Answer 43

So here 3 is same in both the terms. So, we will just equate the powers.

\(2 = \cfrac{3}{2} (4x+6)\)

Is it okay?

Answer 44

no i don't think so

Answer 45

Hmm, where are you confused?

Answer 46

i don't get how the three got eliminated

Answer 47

Well, it is a theorem and its' proof is a little bit higher level.

Answer 48

The theorem says that : if

\(\LARGE{\boxed {\color{Red}{a}^{\color{blue}{m}} = \color{red}{a}^{\color{orange}{n}}}}\)

Then:

\(\LARGE{\boxed {\color{blue}{m} = \color{orange}{n}}}\)

You can say, we didn't eliminate a from there , in fact we just took the powers from there and we are solving that.

Answer 49

(If you have studied logarithm , then this is the proof: )

\(a^m = a^n\)

Taking log both sides :

\(\large{ \sf{ \log {a^m} = \log {a^n} \\
m \log a = n \log a }\\
\textbf{ log a cancels out both sides and we get :} \\
m = n }\)

Answer 50

I know, you have not studies logarithm yet, so it will be very difficult to you to understand that but for now, just understand that when base is same ( in this case , it is 3 ) then the powers are also equal.

Answer 51

Are you still confused?

Answer 52

okay. no not confused

Answer 53

Great . So, you have now :

\(2 = \cfrac{3}{2} (4x+6)\)

Right?

Answer 54

yes

Answer 55

So, isn't it easy to solve for x now? Can you do this Captain?

Answer 56

would you foil?

Answer 57

Yes you can use it here or just to make it easier with the calculation :

\(2 = \cfrac{3}{2} (4x+6)\)

Take 2 common from 4x + 6

\(2 = \cfrac{3}{2} (2(2x+3))\)

Answer 58

So, :

\(\sf{ 2 = \cfrac{3}{\cancel{2}} \times \cancel{2} (2x+3)}\)

Answer 59

So, we get finally :

2 = 3(2x+3)

It will be easier now to find x. Can you do that?

Answer 60

yes

Answer 61

Okay, so, tell me what you get the value for x

Answer 62

x=1.16

Answer 63

*x=-1.16

Answer 64

yes it is correct.

Answer 65

Or you can say : \(x = -\cfrac{7}{6}\)

Answer 66

Thank you!

Answer 67

Thanks for the medal. And Good Luck for your journey on OpenStudy!

Answer 68

Thanks!