INDICES AND LOGARITHMS (INDEKS DAN LOGARITMA)

II. Powers Extended

Ok, we've established that (xy)3 = (xy)(xy)(xy) = xyxyxy.
Multiplication is the same no matter which way you write it (2 * 3 * 4 * 5 = 3 * 2 * 5 * 4), rearranging it gives (xy)3 = xxxyyy.
So, (xy)3 = (x * x * x) * (y * y * y) = x3 * y3

This gives the general rule:

(xy)n = xnyn

This means that a power attached to brackets will affect everything inside the brackets. It doesn't matter how many terms are inside the brackets:

 

Beware though, the above only applies for multiplication/division. Powers do not go well with addition/subtraction of numbers. People like to write (x + y)2 = x2 + y2. Well, it's NOT!

(x + y)2 = x2 + 2xy + y2
(a - b)2 = a2 - 2ab + b2
[These are actually a very useful formulae for dealing with squares. Remember them well.]

It might help to write one hundred times:

I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs. I will not simplify powered brackets that contain plus or minus signs.

Hehehe, just kidding.

 


Powers behave in an interesting way in logarithms. They can become coefficients (pekali)!

log a xn = n log a x

Take note that this can only work if the power is attached to the entire 'subject' of the log. (sorry, I don't know the correct word). The examples below show which types can be converted and which cannot.

log a b2c2 = log a (bc)2 = 2 log a bc

log a xy2 = log a x(y2)
(log a x)2 = (log a x)(log a x)

The first red example cannot be given a single coefficient because x is not squared.
The second red example cannot be given because the entire logarithm is squared: not just the 'subject'.

Read the next page, and come back to have a look at these examples. See if you notice a connection.

 


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