How does 2 ? ( ? + 1 ) + 2 ? ( ? − 1 ) simplify to 2^{1+n}n

[EddyBenzen122]

I had a question in my math textbook that tells me to simplify 2 ? ( ? + 1 ) + 2 ? ( ? − 1 ).
So what I did was simplify 2 ? ( ? + 1 ) to get [math]2^nn+2^n[/math] and [math]2^n(n-1)[/math] to get [math]2^nn - 2^n[/math]then I get [math]2^nn+2^n+2^nn-2^n[/math] where I got stuck not knowing how i can simplify further to 2^{1+n}n

 

[Dr.Peterson]

I had a question in my math textbook that tells me to simplify 2 ? ( ? + 1 ) + 2 ? ( ? − 1 ).
So what I did was simplify 2 ? ( ? + 1 ) to get [math]2^nn+2^n[/math] and [math]2^n(n-1)[/math] to get [math]2^nn - 2^n[/math]then I get [math]2^nn+2^n+2^nn-2^n[/math] where I got stuck not knowing how i can simplify further to 2^{1+n}n

I think the problem must have said to simplify [imath]2^n(n+1)+2^n(n-1)[/imath]. Be careful when pasting, and preview the results.

Do you see that two of the terms in your expansion cancel? Then you can factor out n from the remaining two terms, and use a property of exponents to simplify further.

 

[EddyBenzen122]

I think the problem must have said to simplify [imath]2^n(n+1)+2^n(n-1)[/imath]. Be careful when pasting, and preview the results.

Do you see that two of the terms in your expansion cancel? Then you can factor out n from the remaining two terms, and use a property of exponents to simplify further.

I get [math]n(2^n + 2^n)[/math] after factoring out n and I'm still stuck figuring what I should do with the exponents

 

[canvas]

[math]2^n+2^n=2\cdot2^n=2^1\cdot2^n=2^{n+1}[/math]

 

[pka]

I had a question in my math textbook that tells me to simplify 2 ? ( ? + 1 ) + 2 ? ( ? − 1 ).
So what I did was simplify 2 ? ( ? + 1 ) to get [math]2^nn+2^n[/math] and [math]2^n(n-1)[/math] to get [math]2^nn - 2^n[/math]then I get [math]2^nn+2^n+2^nn-2^n[/math] where I got stuck not knowing how i can simplify further to 2^{1+n}n

[imath]\begin{gathered} {2^n}(n + 1) + {2^n}(n - 1) = {2^n}\left\{ {(n + 1) + (n - 1)} \right\} \\ = {2^n}\left\{ {2n} \right\} \\ = n \cdot {2^{n + 1}} \\ \end{gathered}[/imath]