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

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In summary, the conversation discussed strategies for remembering how to manipulate exponents, specifically in cases where the bases are the same but the exponents differ. Examples using powers of 2 and 10 were given, and it was recommended to practice and internalize the rules to make it easier to solve exponent problems.
  • #1
Tyrion101
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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?
 
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  • #2
[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:
$$2^2 \times 2^3 = 0100000 = 2^5$$
 
Last edited:
  • #3
jbunniii said:
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.
jbunniii said:
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##
 
  • #4
Oops, yes, I left out a couple of zeros! Sorry for the confusion. I'll edit my previous post to fix it.
 
  • #5
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.
 
  • #6
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.
 
  • #7
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.
 
  • #8
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
 

1. How do I know when to add exponents?

To add exponents, the bases of the terms must be the same. Then, simply add the exponents together while keeping the base the same. For example, 23 + 24 = 27.

2. When do I multiply exponents?

Exponents are multiplied when there are multiple instances of the same base. To multiply, keep the base the same and add the exponents together. For example, (23)2 = 26.

3. What if the bases are different when adding exponents?

If the bases are different, the exponents cannot be added. In this case, leave the terms as they are and simplify if possible. For example, 23 + 32 cannot be simplified any further.

4. How can I remember the rules for adding and multiplying exponents?

A helpful mnemonic for remembering the rules is "same base, add the exponents; different bases, leave them be." Another way is to think of the exponents as representing repeated multiplication (e.g. 23 = 2 x 2 x 2) and use that knowledge to determine whether to add or multiply.

5. Can I use the same rules for subtracting and dividing exponents?

No, the rules for subtracting and dividing exponents are different. When dividing, subtract the exponents (e.g. 25 ÷ 23 = 22). When subtracting, the bases must be the same and then the exponents can be subtracted (e.g. 25 - 23 = 22).

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