Hi,

These sort of proofs are done with PMI (Principle of Mathematical Induction).

According to PMI, We can proof this equation in three steps:

- Prove this equation for k=1. LHS=[math](2^1)+(2^1)=2+2=4[/math]. RHS=[math]2(2^1)=2*2=4[/math]
- Prove this equation for k=n (some real number n) . LHS=[math](2^n)+(2^n)[/math] RHS=[math]2(2^n)[/math] , This is already shown in question.
- Prove this equation for k=n+1 (some real number n) . LHS= [math](2^(n+1))+(2^(n+1))=2(2^n)+2(2^n)=2[2^n+2^n][/math]. Using result of step 2 shown above : =[math]2[2(2^n)][/math]. RHS= [math]2(2^(n+1))=2[2(2^n)][/math] .

LHS=RHS. Proved

Hence, by Principle of Mathematical Induction this equation holds true for all real values of k.

Let’s take a step back from your problem, so we can see it more clearly for what it really is. To understand how this equation is true, is to understand what some mathematical “language” means.

Mathematical notationis, firstly, a way of writing out some rather long sentences with many fewer words. It came about because people got tired of saying things like “Think of a number, then multiply two by itself a number of times equal to the number you first thought of, then add the result to itself.” Instead of that sentence, we can just write “k” for “the number you first thought of”, raise2to the powerkin place of “multiply two by itself a number of times equal to”k, and write “+” for “add the result to itself”. Even ourIndo-Arabic number notation, which lets us write “2” for “two”, is just another short-cut, but a very convenient one.And maths notation is more than that – it’s also become an international language for the same ideas. No matter whether you speak English, German, Russian, Chinese or Greek, you can write down the same maths ideas in exactly the same way, yet read them in your own language. How cool is that?

So our first task in solving your maths problem is to understand what this “sentence” is saying. We’ve actually already explained what the stuff before the equals sign “=”, the

left-hand side(orLHS), means. So what does theRHS(right hand side) mean?Whenever we write an expression like

a(b), we’reusing– “(”,parenthesesleft parenthesisand “)”,right parenthesis– which we sometimes loosely call “brackets” –to group stuff together. In the expressiona(b), the parentheses just make sure that we deal with whateverbis before we worry abouta. In your case, you have2^kinside your parentheses, and2before them. We already know that2^kmeans “raise2to the powerk”, or more simply, “multiply2by itselfktimes”.Also, there’s a

convention– an agreed-on meaning – that putting two maths terms side by side (or “juxtaposing” them) – means that we should multiply them. So that an expression likeabcdeis a shortcut for writingaxbxcxdxe. But we have to be careful with terms that include numbers – there’s no point writing “22” to mean “2x2”, since we could confuse that with the usual decimal meaning of “22”, that is “twenty-two”. Still, we can write either “2x2” or “2 (2)” to keep them separate butadjacent(side by side).I’m guessing that you’ve heard of a

rulecalled “BOMDAS” or something similar? This is just another maths convention that helps us ensure that whatever we write down in mathematical language has only one possible meaning – it helps us avoid confusion. For example, when you see the expression2 + 5x6, how do you know whether you should:5to2, then multiply the result by6,or

5by6, then add2to the result?The BOMDAS ruleanswers this question, by giving the order in which we should perform the differentoperations. Those operations, in order, are:Brackets[ … ], braces { …} and parentheses ( … )Of, as we find in expressions like “three-fifths of five-eighths of the total”, or “20 percent of Jack’s wages”MultiplicationDivisionAdditionSubtractionApplying this rule, we see that we have to multiply before adding, so that:

2 + 5x6 = 2 + 30 = 32.In your case, we use the parentheses to keep the first

2separate from the second2, which represents the base2that we raise to the exponentkto get2^k. But the RHS still means that we need to multiply2by2^k.Now that we understand what each side of the equation means, how so we know it’s true? We use the first (secret) super-power of mathematics: our incredible

observationskills! 😉 We notice that on the LHS we have the same term, “2^k”, repeated and added to itself. That’s the same as two lots of “2^k”, which we can write as “2 of 2^k.But what do we do with the word “of” in maths? In the expression “20 percent of Jack’s wages”, we can replace “of” by “multiply” or “x”. Supposing Jack’s wages are $300, then 20% of his ages is 20% x $300 = (20 / 100) x $300 = 0.2 x $300 = $60. And the expression “three-fifths of five-eighths of the total” just means “(3 / 5) x (5 /8) x (the total)”.

So “

2 of 2^k” is exactly the same as “2 x 2^k”, and that’s also the value of the RHS.Now, I’ve taken my time explaining this stuff. You’ll actually need to spend a lot

moretime than I did just now, to make sure you’ve got it. The only reason I understand any of this is because Ihaveput in the time to think about it.One last thing: most of the trouble people have with maths is because nobody ever told them this:

A maths class is a language class!As ever, if anything here is still unclear, feel free to ask more questions -and good luck with your language classes. 😉Let us assume [math]2^k[/math] as a.

Now, the equation becomes a + a which is nothing but 2a.

Let us put back the initial values for a= [math]2^k[/math]

thus, [math]2^k[/math] + [math] 2^k [/math] = 2 ([math]2^k[/math])

Happy Learning!!!

Perhaps the sum of exponential functions leads you to believe that they are special and cannot be treated as regular quantities. But, if we choose to make the substitution [math]u = 2^x[/math], we see that

[math]2^x + 2^x[/math]

[math]= u + u[/math]

[math]= 2u[/math]

[math]= 2 \cdot 2^x[/math]

This is not necessary, however; it should be intuitive that, by the way we define addition and multiplication, adding something to itself is equivalent to multiplying that thing by [math]2[/math]!

[math]2^k + 2^k = 2(2^k)[/math]

let [math]a = 2^k[/math]

[math]a + a = 2(2^k)[/math]

[math]2a = 2(2^k)[/math]

[math]2(2^k) = 2(2^k)[/math] substitute

Simple. Using the left hand side (LHS) of the equation…

[math]LHS = 2^k+2^k[/math]

From here, factorize it.

[math]LHS = 2(2^k) = RHS[/math]

Note: RHS stands for Right Hand Side

You can also do that for RHS.

[math]RHS = 2(2^k)[/math]

From the equation, we know that from the equation there is two of [math]2^k[/math], so it make sense to split them up into this…

[math]RHS = 2^k(1+1) = 2^k 2^k = LHS[/math]

See? Since LHS=RHS, you have proven the statement, [math]2^k+2^k = 2(2^k)[/math].

Let [math]x[/math] be any real number (such as [math]2^k[/math] for real [math]k[/math]), then [math]2x = (1 + 1)x = x + x[/math] by elementary algebra.

May I ask you, what is multiplication exactly?

Multiplication is adding a number to itself a specific number of times.

Here are some examples:

1 + 1 = 2(1) = 2

2 + 2 = 2(2) = 4

3 +3 = 2(3) = 6

4 + 4 = 2(4) = 8

5 + 5 + 5 = 3(5) = 15

Similarly,

2^k + 2^k = 2(2^k)

it can be shown easily by using mathematical induction but let me use simple logic and conclude the same result.

first assuming

k ≠ -∞then divide both by 2^kleft side :2^k / 2^ k + 2^k / 2^ k which leads to1+1 = 2(1)Right side :2 * (2^k / 2^k ) =2(1)Since both sides are equal then

2^k + 2^k = 2 * 2^k.If you have one [math]2^k[/math], and you add another [math]2^k[/math], then all together you have two [math]2^k[/math]’s.

Suppose you have [math]a \in \mathbb{R}[/math]

[math]a + a = 2(a) [/math]

Here we fix k, and let [math]a^{k} = b [/math] then

[math]a^{k} + a^{k} = b+b = 2(b) [/math]

so for any a here if we fix k this a fact that simply adding something here is equivalent to multiplication its repeated addition.

There are few ways to look at this question.

But the easiest way is to make 2^k into x.

Let 2^k be x, x+x = 2x

from here you know x+x is probably equal to 2x.

2x =2x (shown, L.H.S = R.H.S)

Sub x into 2^k, 2(2^k) = 2(2^k).

It would be clear to understand the question by using simple algebra expression which simplified the question. If there is any doubt, feel free to ask me or drop me a message.

Think of it like this:

1(2^k) + 1(2^k) = 2(2^k), because any number is the same as 1 * itself.

Now factor 2^k out to get this:

2^k(1 + 1) = 2(2^k)

Which simplifies to

2(2^k) = 2(2^k)

So, both the expressions are equal, by using a simple proof.

The simplest method possible

2^k=(2^k) x 1= 1(2^k)

substitute this in to the original equation

1(2^k)+1(2^k)

factor out the 2^k

(1+1)(2^k)

1+1=2

substitute

2(2^k)

Anything multiplied by 1 remain same so we can easily write above as :

1 x (2k) + 1 x (2K)

Now since 2K is common in both, we may write it as follows:

2K (1 + 1) or (1+1)2K

Hence,

2(2K)

Proved

The only reason this doesn’t make perfect sense to you is that you are doing something very familiar using unfamiliar symbols. I’m not going to give you a technical explanation. I’m just going to show you how it makes sense.

When you learned how to multiply, you learned that…

4 + 4 = 2 x 4 … or as you are writing it now… 4 + 4 = 2(4). Similarly…

4 + 4 + 4 = 3(4)

4 + 4 + 4 + 4 + 4 + 4 = 6(4)

Now you know that … x + x = 2(x) , or as we write it more simply … 2x

x + x + x = 3(x) or 3x

x + x + x + x + x + x = 6(x) or 6x … and so on.

As long as you are adding the same thing, you can change an addition into a multiplication. So …

(2^k) + (2^k) = 2(2^k)

(2^k) + (2^k) + (2^k) = 3(2^k) and so on…

Well quiet simple (preschool level really):

[math]\Huge{A: 1+1=2}[/math]

Do we all agree on this?

[math]\Huge{B: a+b=c \iff \overbrace{(a+b) \times d}^{=ad+bd}=c \times d}[/math]

Also quiet clear?

Now try to combine [math]\Huge{\text{A and B} \Huge \color{red}{\text{ logical}}}[/math] (hint [math]d=2^k[/math]) and voila, this is how it works.

It’s pretty simple, let’s see. Usually you know you add two or more things that are equal. Like x+x=2x. In this case you are adding two thing leaving it indicated.

[math]2^k+2^k=2(2^k)[/math]

Take LHS, Take [math]2^k[/math] as common factor, we get,

[math]2^k(1+1)[/math]

We know that [math]1+1=2[/math], so, the equation would reduce to,

[math]2^k(2)[/math] or [math]2(2^k)[/math]

I hope my answer was helpful. If you've any questions regarding Mathematics, Physics or Computers, ask me.

Good day my math friend!!!

divide both sides by [math]2^k[/math] and you get the 1 + 1 = 2. The only case when you can not do the divide would be at negative inifinity when the denominator would be 0