How to remember when to add and when to multiply exponents?

[Tyrion101]

I've always had trouble remembering things that are similar, but not the same, like sometimes you add exponents of an expression, is there something I can use to remember this?

 

[jbunniii]

[edit] Fixed the binary representations. My first attempt omitted some zeros.

It might help to think about the special case of powers of 2. For example, $(2^2)(2^3) = (4)(8) = 32$, which equals $2^{2+3}$, not $2^{2 \times 3}$. One way to remember this is to consider the binary representation:
$$2^2 = 0000100, 2^3 = 0001000$$
Multiplication by 2 is equivalent to shifting the representation to the left by one "bit", which adds 1 to the exponent. Multiplication by $2^3$ is the same as multiplying by 2 three times, or equivalently, shifting to the left three times, or adding three to the exponent:
$$2^2 \times 2^3 = 0100000 = 2^5$$

 

[Mark44]

It might help to think about the special case of powers of 2. For example, $(2^2)(2^3) = (4)(8) = 32$, which equals $2^{2+3}$, not $2^{2 \times 3}$. One way to remember this is to consider the binary representation:
$$2^2 = 000010, 2^3 = 000100$$

Correction:
There are too few zeros in the binary representations above, as well as the one later on.
$2^1 = 2 = 000010_2$ This means 1 * 2^1 + 0 * 1.
$2^2 = 4 = 000100_2$ This means 1 * 2^2 + 0 * 2^1 + 0 * 1.
$2^3 = 8 = 001000_2$ This means 1 * 2^3 + 0 * 2^2 + 0 * 2^1 + 0 * 1.

Multiplication by 2 is equivalent to shifting the representation to the left by one "bit", which adds 1 to the exponent. Multiplication by $2^3$ is the same as multiplying by 2 three times, or equivalently, shifting to the left three times, or adding three to the exponent:
$$2^2 \times 2^3 = 010000 = 2^5$$

$2^2 \times 2^3 = 100000_2 = 2^5 = 32$

 

[jbunniii]

Oops, yes, I left out a couple of zeros! Sorry for the confusion. I'll edit my previous post to fix it.

 

[Mark44]

Tyrion, it might helpful to better understand what exponents mean.

Exponents represent repeated multiplication, at least for positive integer exponents, so a² means $a \cdot a$ and a³ means $a \cdot a \cdot a$.

This means we could write (a²)(a³) as $(a \cdot a \cdot a)(a \cdot a)$. We can regroup these factors (associative property of multiplication) as $(a \cdot a \cdot a \cdot a \cdot a)$, or a⁵, since there are 5 factors of a. When you multiply a power of a variable by a power of the same variable, the exponents add.

If we had (a³)², that means (a³)(a³). If you expand each of the two factors as above, you'll see that there are 6 factors of a, so (a³)² = a⁶. When you raise a power of a variable to a power, the exponents multiply.

 

[symbolipoint]

Mark44's post means the most.

Understand the rules of exponents, so you do not need to remember instructions about what to do with the exponents. You should reach the ability to know what to do just by seeing an expression with its exponents. You should also still be able to analyze what you see to enable easier work of simplifications.

 

[jtbell]

When in doubt, work it out like Mark44 did. After doing that enough times, you'll internalize the rules and you'll be able to write down the answer immediately without working out the intermediate steps.

 

[Redbelly98]

I think either working an example using simple numbers (jbunni's 1st suggestion in Post #2) OR working it out with symbols (Mark44, post #3) works best if you are having trouble memorizing the rules. Or like with most things: practice, practice practice.

Just my opinion: the suggestion of using binary representations may not be very helpful to somebody who is having some struggles or trying to wrap their head around exponent manipulation rules. But applying the same logic to powers of ten may work better:

10¹ x 10²
= 10 x 100
= 1,000
= 10³
= 10¹⁺²