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There are two ways to read your question.
(1) If you are asking about typical powers of natural numbers, e.g. squares and cubes of typical numbers, then your best approach is to memorize them. You can always use algebraic techniques to get the answer, but you will be using precious mental resources and time.
(2) If you are asking about the laws of exponents, I ask that my students do not memorize them. I prefer that they understand what is going on and that they can reproduce the laws on their own.
First you must have a basic understanding of the naive definition of exponentiation as repeated multiplication. 2^4=2*2*2*2. Just as the naive understanding of multiplication as repeated addition needs adjusting when you move to decimals, fractions, polynomials, etc... the definition of exponentiation will need to be adjusted. But you need this as a starting point.
(a) Product of Powers: a^m*a^n=a^(m+n) Using the definition of exponentiation we see that 3^2*3^4=(3*3)*(3*3*3*3)=3*3*3*3*3*3=3^6, so you can see that we add exponents.
(b) Power of a Product: (ab)^m=a^m*b^m. (2x)^3=(2x)(2x)(2x)=2*2*2*x*x*x=2^3x^3 again appealing to the definition of exponents.
(c) Power of a Power: (a^m)^n=a^(mn) (3^2)^3=(3^2)(3^2)(3^2)=3^6 using the definition of exponents and the product of powers rule. We could further break this down and only use the definition.
(d) Zero power property: a^0=1 for all a not equal to zero. (Explanation to come.)
(e) Negative exponent rule: a^(-m)=1/(a^m) for a not zero. Consider the following patterns: 4^3=64,4^2=16,4^1=4, 4^0=1,4^(-1)=1/4. As you go from large to smaller powers of 4 you divide by 4 (one reason that 4^0=1) so it makes sense to continue the pattern. Try with other bases. This gives a rationale for a^(-1)=1/a.
(f) Quotient of Powers: a^m/a^n=a^(m-n). So 3^8/3^5=3^3. This comes from the definition, the product of fractions, and finally the fact that a/a=1 for all nonzero a. So we have (3*3*3*3*3*3*3*3)/(3*3*3*3*3) Note that we can pair up 5 threes in the numerator with 5 threes in the denominator; each pair divides to one leaving 3 threes in the numerator.
(e) Power of a Quotient: (a/b)^m=a^m/b^m (3/5)^3=3/5*3/5*3/5; using fraction multiplication and the definition of exponents gives 3^3/5^3
The rules reinforce one another: a^m*a^(-m)=1: we can add the exponents to get an exponent of zero and apply the zero exponent rule, we could rewrite as a^m*1/a^m and note that the numerator divides out with the denominator, etc...
You should be able to reproduce the laws and know, for example, when to add and when to multiply exponents by doing simple examples as needed.
A reminder that exponents are a shortcut for multiplication.
There is an easy way to remember exponents if you know the previous exponent. This only works for squares though.
So, if you know n², you can find (n+1)² using this method.
By expanding (n+1)², you'll arrive at n² + 2n + 1. You can use this equation to find the square of your next base.
example: you know that 11²=121.
Say you what to find 12². Simply plug 11 into the equation.
this equation is very flexible. Say you know 11², but you want to find 13².
just change the equation to (n+2)², which expands to n²+4n+4. Plug and chug.
I recommend you memorize the basic squares from 1-15, and then you can use this equation for higher numbers.
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