If $2^{2013}-2^{2012}-2^{2011}+2^{2010} = k \cdot 2^{2010}$, find $k$

Question

Hey this is just a question i was having fun with but couldnt solve for some reason. Would love if you can help me solve it thankyou!:

If $2^{2013}-2^{2012}-2^{2011}+2^{2010} = k\times2^{2010}$. find k

Answer 1

Well there is of course the option of simplifying via a calculator but if you want to do it by hand, first I would pull out the factor of $2^{2010}$ out of the left hand side to get $$2^{2010}(2^3-2^2-2+1)=k\cdot 2^{2010}$$ Then simplify and divide both sides by $2^{2010}$ to get $$\frac{2^{2010}\cdot 3}{2^{2010}}=k$$ Which yields $$3=k$$

Answer 2

Let $a=2^{2010}$ and just bring down some of the exponents: $$2^{2013}-2^{2012}-2^{2011}+2^{2011} = 8a - 4a -2a +a$$ $$= 3a = 3\times 2^{2010}$$

So $k=3$.

Answer 3

The knowledge you want to use:

$$x^{a+b}=x^ax^b$$

Hence you should be able to write everything in multiple of $2^{2010}$, by manipulating the coefficients to get $k$

Solution:

$$2^{2013} - 2^{2012} - 2^{2011} + 2^{2011}\\ =2^3 \cdot 2^{2010} - 2^2 \cdot 2^{2010} =6 \cdot 2^{2010}$$

Answer 4

We have

$$2^{x+2} - 2^{x+1} + 2^{x} = k\cdot 2^{x}$$

therefore

$$k = 2^{2} - 2^{1} + 1 = 3$$

Answer 5

As originally written, the answer was 4

2^2013 -2^2012 -2^2011 +2^2011 = k * 2^2010

2^2013 -2^2012 (-2^2011 +2^2011) = k * 2^2010

2^2013 -2^2012 +0 = k * 2^2010

(2^2010 * 2^3) - (2^2010 * 2^2) = k * 2^2010

2^2010 (2^3 - 2^2)
------------------------- = k
2^2010

2^3 - 2^2 = k

8 - 4 = k

4 = k

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Now, by the same reasoning, the answer becomes

8-4-2+1 = k

3 = k