What is an easy way to remember the laws of exponents? I remember what they are when we go through them, but I have trouble remembering what law applies to a certain question if I'm presented that question with no walk-through.
Exponents actually make it easier to solve complicated problems. However, because they have rules to follow, they do sometimes become confusing. The main things to remember are:
- Bases must be the same if you want to add or subtract powers. `2^(2x) times 4^(2x)` first need the same base `= 2^(2x) times (2^2)^(2x)` . `therefore =2^(2x+4x)= 2^(6x)`
- Bases must be multiplication and division problems to add or subtract powers so do not use the rules for problems (explained later) such as `3^(3x) +3^((3x+2))`
One difficulty is often whether to add or whether to multiply exponents. The rules says `(b^m)^n = b^(m times n)` and `b^m times b^n=b^(m+n)` where b stands for base (b=base) and m and n represent the powers so you could use any variable such as x, y, a and so on. To remember it, think what it actually means.
To make it easier, let's use an example of b=x and m=2 and n=3. So for the above rules we may want to compare `(x^2)^3` and `x^2 times x^3` . What does this mean:
`(x^2)^3= x^2 times x^2 times x^2` which is `x times x timesx timesx timesx times x` and from that we can see that there are actually 6 of the x-es so we can safely say
and that proves that we must multiply when the power is immediately outside the bracket. Now for
`x^2 times x^3 = x times x times x times x times x` and from that we see that we have 5 x-es so we can safely say that `= x^5`
Similarly for division: `b^m/b^n` `= b^((m-n))` and if we use b=x, m=2 and n=3 as above we get `x^2/x^3=` `x^((2-3))= x^(-1)` and as negative powers do not mean minus but divide we move the x and its power to the bottom. This can be confusing but if we picture it, it makes sense:
`therefore =x^2/x^3 = (x.x)/(x.x.x)` from which we can see `= (1.1)/(1.1.x)` because `x ` goes into itself once `= 1/x`
So basically, do not be fazed by exponents, just count them if there is any confusion.
Another problem with exponents is that we forget to apply the normal rules of factorizing etc thinking that, because exponents have their own rules, the rules we learnt before do not apply. The rules do apply, we just have to use them with the exponential rules. So, to compare a question which says, Simplify `2x^2 + 8x` , we would factorize that because we do not have like terms to add. We would never add `2x^2` to `x` because they are not like terms `therefore= 2x(x+4)` .
Same for exponents. If a question says `2^(3x+2)+ 8^x` , we use the factorizing rules by finding a common factor because, like standard algebra (above) we remember that we cannot add non-like terms and adding and minusing exponents applies to multiplying and dividing bases.
`therefore = 2^(3x+2) + (2^3)^x = 2^(3x) (2^2+1)` which `= 2^(3x).(5)` or `5.2^(3x)`
and to simplify the `3^(3x) + 3^((3x+2))` from earlier. Try it yourself and then check here for the answer:
`= 3^(3x)(1+3^2) = 3^(3x)(1+9)=10.3^(3x)` .
To recap always consider these points and have confidence:
- Exponents work with same bases in applying the rules
- Remember anything to the power of 0=1 (`x^0=1` , `5^0=1` , and so on and just to take care as to which part of the term or expression is to the power of 0 take care, for example, `2(x^2)^0=2 times 1=2`
- When the bases are multiplied, the exponents are added provided that the bases are the same.
- Use the factorizing rules when terms themselves are added or subtracted (remember` 3^(3x) +` ....something will require factorizing. Powers cannot be added or subtracted unless the expression is times or divide.)
Hope it helps.
I tell my students to remember MaDsPm - "you get the mads at night"
When Multiplying, Add the exponents
When Dividing, Subtract the exponents
When rasing a power to a Power, Multiply the exponents