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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?
Correction: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:
$$2^2 = 000010, 2^3 = 000100$$
##2^2 \times 2^3 = 100000_2 = 2^5 = 32##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:
$$2^2 \times 2^3 = 010000 = 2^5$$
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.
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.
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.
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.
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).