Solving 3 x √27 = 3^N to Finding the Value of N

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    Calc 3
In summary, the conversation is about finding the value of N in the equation 3 x √27 = 3^N. The conversation provides various hints and suggestions, such as rewriting the equation to make it easier to solve and using index laws. The final solution is N=2.5.
  • #1
thomas49th
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Hi how do I go about finding out what N is to
3 x √27 = 3^N

this can be written as

3 x (√3 x √3 x √3)
which makes
9√3 = 3^N

now where though

i presume n must be a fraction somthing like x/2 ? x is a number? Is this right. How do i go about finding the answer?
 
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  • #2
Of only you could rewrite the equation as 3^{something} = 3^N, it would be obvious...
 
  • #3
huh? Sorry I don't see...was that a suggestion/hint or comment

as I've got a root it does the n have a 2 as it's denominator?
 
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  • #4
An eager student would take the logs of both sides. A clever student would notice something special about this question enabling him to skip taking logs and immediately solve an equation which would give him his answer.

If you're not that clever student, take logs. Solve from there. Then see if you can see what the clever student would do.
 
  • #5
GCSE students don't learn logs. This is on a non-calculator paper. I can't remeber what to do. I've looked in my textbook, revision book, bitesize, s-cool.co.uk and can't see somthing to help... so I turn here. Can someone show me the steps or continue my work so far? I'm doing a pratice paper and their designed to show you what you don't know... I searched books and web and had no luck :(
 
  • #6
Since you can have LHS in terms of [itex]\ \sqrt{3}[/itex]why not try expressing RHS as [itex] \ ( \sqrt{3} )^{something} [/itex]
 
  • #7
thomas49th said:
GCSE students don't learn logs. This is on a non-calculator paper. I can't remeber what to do. I've looked in my textbook, revision book, bitesize, s-cool.co.uk and can't see somthing to help... so I turn here. Can someone show me the steps or continue my work so far? I'm doing a pratice paper and their designed to show you what you don't know... I searched books and web and had no luck :(

Well, you're forced to be the clever student then, aren't you! Hurkyl's post was a hint. To put it more blatantly-- try writing the left hand side as 3 to the power of something. Can you write sqrt(3) as 3^x, for some x? Can 9 be expressed as 3 to some power? Finally, can you express a product of two numbers with the same base as one number?
 
  • #8
Interestingly enough you could write it 9√3 OR you could write it as √81 x √3... Try going on from that approach. Oh, and remember √x = x^(1/2).
 
  • #9
I have no idea how Feldoh's first comment helps, but yes his second does. BIG Hint. Have you learned your Index Laws?
 
  • #10
If you haven't got it yet, I'll give you a more explicit hint. x->3^x is a well defined map, ie, if x=y then 3^x=3^y.
 
  • #11
Finally, if you still don't understand,
[tex]\sqrt{a}= a^{\frac{1}{2}}[/tex].
 
  • #12
Even More Finally if you still don't get it, give up.

N=2.5
 
  • #13
I still don't see...

x->3^x is a well defined map,

I don't see what that means...

I know that [tex]\sqrt{a}= a^{\frac{1}{2}}[/tex] but because I have a whole number and a surd I am not sure what to do. Can someone just WRITE OUT the steps they did one by one using latex?

Thx
 
  • #14
[tex]3\sqrt{27} = 3^n[/tex]
[tex]\sqrt{27}=3\sqrt{3}[/tex]
Therefore:
[tex]9\sqrt{3}=3^n[/tex]
[tex]3^2 \cdot 3^{\frac{1}{2}} = 3^n[/tex]
Since [tex]a^m \cdot a^n = a^{m+n}[/tex], then
[tex]3^2 \cdot 3^{\frac{1}{2}} = 3^{2.5}[/tex]
[tex]3^{2.5} = 3^n[/tex]
[tex]n=2.5[/tex]
 
  • #15
NICE! Yeh I see now...ingenius.
Thanks for all the help :redface:
 
  • #16
No, not ingenious- just basic formulas. Long before you attempt a problem like this you should have learned
[tex]a^x a^y= a^{x+ y}[/tex]
and
[tex](a^x)^y= a^{xy}[/tex]

(I am not saying that Gib Z isn't ingenious- just that it wasn't needed here!)
 

Related to Solving 3 x √27 = 3^N to Finding the Value of N

What is Non Calc 3 x √27 = 3^N?

Non Calc 3 x √27 = 3^N is an algebraic equation that involves multiplication, square roots, and exponents. It is a mathematical expression that represents the relationship between the number 3 and an unknown number, represented by the variable N.

What is the purpose of Non Calc 3 x √27 = 3^N?

The purpose of Non Calc 3 x √27 = 3^N is to solve for the unknown variable N. By using algebraic techniques, we can manipulate the equation to isolate the value of N and determine its numerical value.

How do you solve Non Calc 3 x √27 = 3^N?

To solve Non Calc 3 x √27 = 3^N, we can first simplify the expression by evaluating the square root of 27, which is 3. This gives us the equation 3 x 3 = 3^N. Then, we can use the properties of exponents to rewrite the equation as 3^2 = 3^N. Since the bases are the same, we can set the exponents equal to each other, giving us the solution N = 2.

What are the key steps in solving Non Calc 3 x √27 = 3^N?

The key steps in solving Non Calc 3 x √27 = 3^N are simplifying the expression, using the properties of exponents, and setting the bases equal to each other to solve for the unknown variable N. It is also important to check the solution by plugging it back into the original equation to ensure it is correct.

What are some real-world applications of Non Calc 3 x √27 = 3^N?

Non Calc 3 x √27 = 3^N can be used in various real-world scenarios, such as calculating compound interest, population growth, or radioactive decay. It can also be applied in engineering and physics to solve for unknown quantities in mathematical models and equations.

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