# adding exponents with unknowns with the same base

How do you add numbers with the same base but with unknown exponents? For example, $$2^{x+2} + 2^{x+2}$$ I understand that you do $$2^x \cdot 2^2$$ but I get stuck here. I don't know what to do from here or where to get the working out for the next step.

Thanks

• Is that $2^{x+2}$ or $2^x + 2$? Sep 3 '20 at 6:28
• @Ben Grossmann the first one. Sep 3 '20 at 6:36
• So in general if you want to calculate $$n^a + n^b$$ there is no simplified form for this. It's only if you have multiplication that you can simplify these, for example $$n^a \cdot n^b$$ This you could simplify to $n^{a+b}$. Sep 3 '20 at 7:09

An expression of the form $$a^n + a^m$$ is already in simplest form.

You may, however, if you wish: decrement an exponent by factoring out an $$a$$ from the term, or increment an exponent by factoring out $$\frac1 a$$ from the term. That is, the above expression can be written equivalently as any of the following:

$$(a)(a^{(n-1)}) + (a)(a^{(m-1)})$$

$$\frac{a^{(n+1)}} {a} + \bigl( \frac{1} {a} \bigr)(a^{(m+1)})$$


By factoring out $$a$$ or $$\frac1 a$$ more than once, you can write the equation as:

$$(a^3)(a^{(n-3)}) + \bigl( \frac {a^{(m+5)})} {a^5} \bigr)$$

which we can tell is equivalent to the original expression by using exponent rules:
$$(a^3)(a^{(n-3)}) + \bigl( \frac {a^{(m+5)})} {a^5} \bigr) = a^{ ( n-3 ) +3 } + \bigl( \frac{1} {a^5} \bigr) \bigl( a^{(m+5)} \bigr) = a^n + ( a^{-5} ) ( a^{(m+5)} ) = a^n + a^{ ( m+5 ) -5 } = a^n + a^m$$


One last thing worth mentioning is that you may also factor out common multiples between the terms. Example:

$$a^n + a^m = ( a^4 ) ( a^{n-4} ) + ( a^7 ) ( a^{m-7} ) = a^4 \Bigl( ( a^{n-4} ) + (a^3) ( a^{m-7} ) \Bigr)$$


Ultimately, while re-writing equations in more complicated forms such as in the manner shown here is not usually useful on its own, doing so $$\mathit is$$ often useful when working with more complex equations or in writing proofs.
If this response has been insufficient in answering your question, I'd recommend re-visiting the basic rules of exponents, starting with how addition and multiplication are related, and then moving to how multiplication and exponents are related.
Best regards~

I will assume you are talking about adding terms which have a constant base and some linear monic expression in the exponent.

You can do:

$$2^{x+2} + 2^{x+2}$$

= $$2\cdot2^{x+2}$$

= $$2^1\cdot2^{x+2}$$

= $$2^{1+(x+2)}$$

= $$2^{x+3}$$

Using these properties.

• The original was not 2^x + 2 i meant 2^(x+2) where (x+2) is the exponent. Sep 3 '20 at 10:14
• did you mean $2^x + 2^{x+2}$ ? Sep 3 '20 at 10:16
• No, they both should be 2^(x+2) Sep 3 '20 at 10:34
• If so your question needs an edit. Sep 3 '20 at 10:46
• I had it like so but someone edited it wrong Sep 3 '20 at 10:50