No idea -- algebraic manipulation involving powers

[Chaos_Enlightened]

Homework Statement

If x=2+2^1/2+2^2/3. Then x^3-6x^2+6x=?

Homework Equations

(A+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca+)

The Attempt at a Solution

x^2= 6+3(2^2/3)+2^4/3+4(2^1/3)[/B]

 

[Chaos_Enlightened]

Should I go on and find x^3

 

[Sahil Kukreja]

Hint :- take '2' to the left side and then cube both sides.

 

[Chaos_Enlightened]

Ok

 

[Chaos_Enlightened]

Got x^3-6x^2+12x=14+6(2^1/3+2^2/3)

 

[Sahil Kukreja]

Got x^3-6x^2+12x=14+6(2^1/3+2^2/3)

I think there is a calculation mistake here, sorry if i am wrong.

 

[Nidum]

Work out the numerical value of x from the first equation .

What do you do then ?

 

[Sahil Kukreja]

Hint : work out the numerical value of x from the first equation .

What do you do then ?

The question is solvable without calculating square roots and cube roots.

 

[Delta2]

factorize the polynomial, if all go well (i.e the polynomial has 3 real roots) you should get it in the form $(x-r_1)(x-r_2)(x-r_3)$

You can easily see that one root of the polynomial is...

 

[Nidum]

Think about what sort of answer the question really requires .

 

[Chaos_Enlightened]

That might just do it

 

[Chaos_Enlightened]

Which one should I factorise

 

[Delta2]

$x^3-6x^2+6x=x(x^2-...+...)$ the polynomial inside the parenthesis is a 2nd order polynomial i believe you know how to factorize it,

 

[Chaos_Enlightened]

Wait
I could just substitute then
X^3-6x^2+12x=14+6(2^1/3+2^2/3). ==:
x^3-6x^2+12x=14+6x-12. ==:
Ans. Is 2

 

[Chaos_Enlightened]

Thank You All!

 

[Sahil Kukreja]

Your question is wrong. it should be :- x= 2 + 2^(1/3) + 2^(2/3)

otherwise your calculations are wrong, you can just check.

 

[SammyS]

Homework Statement

If x=2+2^1/2+2^2/3. Then x^3-6x^2+6x=?

Hello Chaos_Enlightened. Welcome to PF !

Your original expression, has some ambiguity in it.

Using the standard Order of Operations, What you wrote, x=2+2^1/2+2^2/3 , literally means $\ x=2+2^1/2+2^2/3\ .$ ^†

Use parentheses around the fractional exponents as in x=2+2^(1/2)+2^(2/3), which gives $\ x=2+2^{1/2}+2^{2/3}\ .$ ^†

You may find the superscript feature (X² button) helpful as well. Using this gives: x=2+2^1/2+2^2/3.

^† LaTeX was used for these.

 

[Chaos_Enlightened]

Thank you Sammy's

 

[late347]

$x^3-6x^2+6x=x(x^2-...+...)$ the polynomial inside the parenthesis is a 2nd order polynomial i believe you know how to factorize it,

I don't think that it factorizes any more than that at least in integers. I could be wrong though it would not be the first time...

I think it seems that the first common and therefore true factor is the lonely (x) (something...)

(x) (x² -6x +6)

closest I managed to get was (x-3)(x-3) which yields out too much stuff. x^2 -6x + 9

(x-2) (x-3) yields too little stuff x^2 -5x + 6

x* x * (x-6 + 6/x) = (x^3) - (6x^2) +( 6x)

 

[late347]

for x² -6x +6 = 0

x is either (6+ √12) / 2
or
x is (6-√12) / 2

Its not very easy to calculate the original equation when the tip is used to switch 2 to the left side and cube both sides. Newton's binomial formula or pascals triangle can be used here. Maybe I am on the wrong track but looking at the calculation it will not be easy or fast to compute those roots into cubes. (X-2)^3 will be easier but also quite tedious.
x= 2 + 2^0.5 + 2^2/3
(x-2)³ = [(√2) + ³√4) ] ³(x-2)^3 = [x+(-2)]³
calculate with binomial theorem or Newton's binomial formula which is found in some mathematics formula books ( It seems I used simplified version from wikipedia https://en.wikipedia.org/wiki/Binomial_theorem scroll down in the article until there is (x+y)³). At least the factors themselves can be seen from pascal's triangle easily. In this case the factors are 1,3,3, and 1.

x³-6x² +12x -8

I'm sure that the right hand side yields something even uglier with pen-and--paper calculation...

 

[late347]

I think I got the answer by looking at what I got from the binomial formula.

• I recommend you start from the left side first (x-2)³

• I tend to agree with Sahil Kukreja's advice. He or she provided a good tip. ( I think ) .
• I think I got the basic idea, in order to get the "rough answer" in unfinished form. It should still be mathematically correct but your teacher will likely have the finished form answer in the answerbook
• try to get the "rough form answer" first, then worry later about expanding the (squareroot + cubic root)³

• Mathematically it's not very significant - it's more of a style issue, I suppose.

(x-2) ³ = (2^0.5 + 2^2/3) ³

. In the end phase we must give the answer for the question "What does this clause yield: clause is; x³ -6x² + 6x"
The answer must be another clause, which equals that quadratic clause. The full clause in the ending (both sides of equation) will be an equation, in that sense. So, therefore the ending clause should be properly balanced like all equations, if that makes it clear. This is my interpretation at least.
But the question asks simply (what does the left side clause give you, on the right side clause?)

• We know for a fact that this equation holds true

x=2 + √2 + ³√4

• Therefore

(x-2)³ = [√2 + ³√4] ³ This equation must hold true as well (it f*****g should hold true!)

• Therefore

What does the clause X³ -6X² +6X equal to?
Give the clause, which equals to that earlier clause. [ X³ -6X² +6X ]


to-do-list

1. first equation was x= 2 + 2^0.5 + 2^2/3
2. in the first equation, minus 2 from both sides.
3. in the first equation, then cube both sides (third power both sides)
4. calculate the (x-2)³ with Newton's binomial formula, or use the wikipedia article about binomial theorem, and find the formula for (x+y)³. Use pen-and-paper and be careful with calculation!

5. according to my limited knowledge, you can substitute the negative sign, simply by using brackets and minus inside
(x-2)^3 = [x+(-2)] ^3

When you calculate
a+(-b)
it is the same as
a-b

At least I hope so because otherwise I screwed up my own solution hehe...

 

[late347]

It appears my method yields correct result. I did the rough answer first on pen-and-paper.

Then I plugged my paper results into the calculator and verified the equation holds true on both sides.
I think that it may be unnecessary to expand the (2^0.5+2^2/3) ³, especially if you have calculator available to verify.

But one method is to use the binomial formula for (x+y)³. I can't think of any easier way to tackle this problem then simply expand that one out... until you have some result according to the formula.

x³ -6x² + 6x = ?

Calculator can be used at the end to verify that both sides of equation hold true. That's what I would do, but I could be wrong though.

 

[SammyS]

Wait
I could just substitute then
x^3-6x^2+12x=14+6(2^1/3+2^2/3). ==:
x^3-6x^2+12x=14+6x-12. ==:
Ans. Is 2

Where did you get $\ 6\left(2^{1/3}+2^{2/3}\right) \ ?$

... or should the original problem state that $\ x=2+2^{1/3}+2^{2/3}\$ rather than $\ x=2+2^{1/2}+2^{2/3}\ ?$

 

[Sahil Kukreja]

Where did you get $\ 6\left(2^{1/3}+2^{2/3}\right) \ ?$

... or should the original problem state that $\ x=2+2^{1/3}+2^{2/3}\$ rather than $\ x=2+2^{1/2}+2^{2/3}\ ?$

Yes, that's what i was saying in post #16.

 

[SammyS]

Yes, that's what i was saying in post #16.

Post #16:

Your question is wrong. it should be :- x= 2 + 2^(1/3) + 2^(2/3)

otherwise your calculations are wrong, you can just check.

I see that I missed that post of yours.

Then it's questionable as to whether OP will return to tell us. He's had over a day to respond to your post.

Certainly, $\ x=2+2^{1/3}+2^{2/3}\$ is much easier to work with. However, I was surprised that the problem as posted (likely with a typo) gave a fairly simple result, having only two terms.

Spoiler:

$\ 6\cdot 2^{5/6}-4\cdot2^{1/2}=5.033\dots\$

 

[late347]

Where did you get $\ 6\left(2^{1/3}+2^{2/3}\right) \ ?$

... or should the original problem state that $\ x=2+2^{1/3}+2^{2/3}\$ rather than $\ x=2+2^{1/2}+2^{2/3}\ ?$

I think the most important thing in the beginning is to utilize the original equation

X= 2+ (2^0.5 ) + (2^(2/3))

Work your way from the original equation until you reach on one side of the equation this clause below;

(x^3) - (6x^2) +(6x) = ?

The fastest way to reach into this clause from the original equation is to subtract 2 from both sides.

Then you raise both sides of equation to the third power.

 

[late347]

my solution in spoiler

question stated such things as below...

• when x= 2 + [2^(0.5)] + [2^(2/3)]

• What then, if anything, does x³-6x²+6x equal to?

Spoiler: rough answer to the problem

(x-2)^3 = (√2 + ³√4) ^3

We use binomial formula to expand the lefthand side out of the brackets

x³-6x²+12x-8 = (√2 + ³√4) ^3 ]] both sides +8

x³-6x²+12x= (√2 + ³√4) ^3 +8 ]] both sides -6x

x³-6x²+6x = (√2 + ³√4)³ +8 -6xx³-6x²+6x= (√2 + ³√4)³ +8 -6*(2 + √2 + ³√4)

"The very thing it was required to have shown."