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

  • Thread starter Thread starter Tyrion101
  • Start date Start date
  • Tags Tags
    Exponents
AI Thread Summary
Understanding when to add or multiply exponents hinges on recognizing the rules of exponents. When multiplying like bases, such as in (a^2)(a^3), the exponents add, resulting in a^5. Conversely, raising a power to another power, like (a^3)^2, multiplies the exponents, yielding a^6. Visualizing this with binary representations can aid in grasping the concept, particularly with powers of 2, where multiplication corresponds to shifting left in binary form. Ultimately, consistent practice and application of these rules will lead to better intuition and understanding of exponent manipulation.
Tyrion101
Messages
166
Reaction score
2
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?
 
Mathematics news on Phys.org
[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:
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##
 
Oops, yes, I left out a couple of zeros! Sorry for the confusion. I'll edit my previous post to fix it.
 
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.
 
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.
 
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.
 
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
 
Back
Top