Powers function n^5+n^4=(n^5−n^3)(n−1)−(n−2)

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In summary: So I guess it's just a way of thinking about the problem.In summary, the conversation discusses two equations and their solutions when n is replaced by any number greater than 0. The first equation is not an identity and requires root finding techniques to find the values of n for which it is true. The second equation is an identity with the additional restriction that n cannot equal 1. The conversation also explores a philosophical question about a 3 digit number that has the same number in every digit, and when you add the digits and multiply by 37, you get the same number again. The hint given is that 3 times 37 equals 111, which can be used to explain this pattern.
  • #1
Angel11
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Hello again,i am back. So now i have diffrent questions while spending my time in powers for some reason that i can't explain...Anyways let's get to the point. these ones are really simular to the previous one.
n^{5}+n^{4}=(n^{5}-n^{3})*(n-1)-(n-2), n^{4}+n{3}=(n^{4}-n^{2}*(n/(n-1)). You can try to replace n by any number which is >0. And my question is:"WHY does this happen (i mean it goes a little bit into philosophical thinking rather than logical but i would like an explanation like why and 3 digit number that has the same number in every digit if you add it together and multiply the result by 37 it becomes the same number).(Whew) Also if you had the patience to read all of this and acctoully came up with an answer i want to say you are a legend (Yes)
 
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  • #2
Hint: $ 3 \cdot 37 = 111 $
 
  • #3
Let's look at your first equation:

\(\displaystyle n^5+n^4=\left(n^5-n^3\right)(n-1)-(n-2)\)

If we use $n=1$, we get:

\(\displaystyle 2=1\)

So, it's not an identity...and we would have to use a root finding technique to get approximations for the real values of $n$ for which the equation is true:

\(\displaystyle n\approx0.88157969798488504310\)

\(\displaystyle n\approx2.6222866129011998799\)

Let's look at the second equation you gave (I assume it is the following):

\(\displaystyle n^4+n^3=\left(n^4-n^2\right)\left(\frac{n}{n-1}\right)\)

\(\displaystyle n^4+n^3=\frac{n^5-n^3}{n-1}\)

Since you stated $0<n$ we may divide through by $n^3$ to obtain:

\(\displaystyle n+1=\frac{n^2-1}{n-1}\)

Multiply through by $n-1$:

\(\displaystyle (n+1)(n-1)=n^2-1\)

So, given that this is an identity, you original equation is as well, with the additional restriction $n\ne1$. :)
 
  • #4
Alternatively,

$$ \left( n^4 - n^2 \right) \left( \frac{n}{n - 1} \right), \, n \gt 0, \, n \ne 1 $$

$$ = \frac{ n^5 - n^3 }{ n - 1 } $$

$$ = \frac{ n^3(n^2 - 1) }{ n - 1 } $$

$$ = \frac{ n^3(n + 1)(n - 1) }{ n - 1} $$

$$ = n^3(n + 1) $$

$$ = n^4 + n^3 $$
 
  • #5
Angel1 said:
\(\displaystyle n^{5}+n^{4}=(n^{5}-n^{3})*(n-1)-(n-2)\)
I suspect that it is supposed to be
\(\displaystyle n^5 + n^4 = (n^5 - n^3) \left ( \frac{n - 1}{n - 2} \right )\)

similar to the other one. However now we have no solutions for n, integer or otherwise.

-Dan

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greg1313 said:
Hint: $ 3 \cdot 37 = 111 $
Okay I'm officially clueless, a state which I have a lot of experience in. How is this a hint?

-Dan
 
  • #6
topsquark said:
Okay I'm officially clueless, a state which I have a lot of experience in. How is this a hint?

-Dan
The original question was ":"WHY does this happen (i mean it goes a little bit into philosophical thinking rather than logical but i would like an explanation like why and 3 digit number that has the same number in every digit if you add it together and multiply the result by 37 it becomes the same number)."
For example, if we add the digits in 555 we get 15 and 555/37= 15. If we add the digits in 888 we get 24 and 888/37= 24.

That is, "why, if we have something like aaa, where "a" is a single digit, if you add them, you get "3a", while if you divide by 37, you also get "3a"". And greg1313's response was that aaa= a(111)= a(3)(37).
 

FAQ: Powers function n^5+n^4=(n^5−n^3)(n−1)−(n−2)

What is the Powers function?

The Powers function is a mathematical expression that follows the form n^5+n^4=(n^5−n^3)(n−1)−(n−2). It is a polynomial function that involves raising a number to the fifth and fourth power.

What is the significance of the number 5 in the Powers function?

The number 5 in the Powers function represents the degree of the polynomial. In this case, it is a fifth degree polynomial because the highest exponent is 5.

What is the purpose of the Powers function?

The Powers function can be used to solve various mathematical problems and equations. It is also used in many fields of science, including physics, engineering, and economics.

How do I solve the Powers function?

To solve the Powers function, you can use algebraic methods such as factoring, simplifying, and substitution. You can also use a graphing calculator to find the roots or solutions of the equation.

What is the relationship between the two sides of the equation in the Powers function?

The two sides of the equation in the Powers function are equal. This means that for any given value of n, the expressions on both sides will result in the same number. This is known as the equality property of equations.

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