The Mathematis 11 Competey Test Laws of Expoets (i) multipliatio of two powers: multiply by five times 3 x = ( x x ) x ( x x x x ) = 8 multiply by three times et effet is to multiply with a total of 3 + = 8 times Thus = = 3 3+ 8 I symbols, we a write that if is ay umber, the i = + m m The illustratio above whih shows why multiplyig two expoetials together gives a ew expoetial whose expoet is the sum of the origial expoets a learly be exteded to produts of three or more expoetials with the same base. The total umber of fators i the produt is equal to the sum of the fators from all expoetials ivolved, so the expoet i the simplified produt will be just the sum of the expoets i the fators. This is illustrated i the seod example below. examples: i = = + 11 i i i = = 3 8 + + 3+ 8 18 (ii) divisio of oe power by aother: multiply by seve times = x x x x x x x x x = x x 1 = 3 multiply by four times Here, the four fators of i the deomiator ael four of the fators of i the umerator, leavig a et of three fators of i the umerator. The deomiator of 1 a simply be dropped to get the fial result 3 overall. Notie that this simplifiatio a be writte more ompatly as David W. Sabo (003) Laws of Expoets Page 1 of
= = 3 sie if we are outig up overall fators of i the expressio, the umber of fators of i the deomiator must be subtrated from the umber of fators of i the umerator. I symbols, if is ay ozero umber, ad m is a larger umber tha, we a write m = m Note that if we started with the the deomiator has more fators of tha does the umerator. Whe all possible aellatio of fators is doe, there will be three fators of left o the bottom, ad oe o the top: 1 1 = = 3 So, i symbols, if is ay ozero umber, but ow > m, we get m = 1 m examples: 8 = = 8 1 1 = = 8 8 + 8 i = = = = 3 3 3 8 3 or i 3 = = + 3 From this last example, you a see that if two or more powers with the same base are multiplied i the umerator or the deomiator or both, the the fial result will have a power equal to the sum of all expoets i the umerator mius the sum of all expoets i the deomiator. This oly works for those powers that have the same base. David W. Sabo (003) Laws of Expoets Page of
(iii) raisig a power to a power: multiply by three times ( ) 3 = ( ) x ( ) x ( ) = ( x ) x ( x ) x ( x ) sie = x = x x x x x = six fators of This amouts to otig that ( ) 3 = = 3 I symbols, if is ay umber, the ( ) m = = m m I the last form i the box, we have used the algebrai ovetio that the produt x m a be writte simply as m. example: ( 3 ) = 3 = 3 8 ( whereas 3 i 3 = 3 + = 3 ) To summarize so far: Whe a power is raised to a power you multiply the two expoets together. Whe a power is multiplied by aother power with the same base, you add the expoets. Whe a power is divided by aother power with the same base, you subtrat the seod expoet from the first. David W. Sabo (003) Laws of Expoets Page 3 of
(iv) raisig a produt to a power: x 3 multiplied four times ( x 3) = ( x 3) x ( x 3) x ( x 3) x ( x 3) = x 3 = 3 four fators of 3 ad four fators of I geeral, the, if ad d are ay umbers, example: ( ) d = d ( ) 3 = 3 (v) raisig a quotiet or a fratio to a power: If is ay umber, ad d is ay ozero umber, the = d d So, for example 3 3 3 3 3 3 3 3 3 3 3 3 = = = (vi) A Cautio! The five laws of expoets summarized i the square boxes above all ivolve multipliatio or divisio oly. As soo as additio or subtratio ours, the obvious relatios are NOT TRUE! For example, But, ( ) + 3 = = ( ) + 3 + 3, sie this latter expressio evaluates to 1 + 81 = 9, whih is learly iorret. Aother example is David W. Sabo (003) Laws of Expoets Page of
but ( ) = 3 = 3 ( ) sie the latter gives 180 10 = 183, whih is learly wrog. I both of these ases, the rules of priority for operatios (desribed just ahead i these topi otes) are followed i the orret forms. It is eessary to evaluate the quatity i brakets first, ad the apply the expoet to the result. The expoets here apply to the result of doig whatever operatios are show iside the brakets. This will beome a very importat rule to remember whe the brakets otai symboli expressios rather tha simple umerial expressios. So, for powers of sums ad differees, there is o simple law (suh as forms (iv) ad (v) above for powers of produts ad quotiets, respetively) that allows us to express them i terms of powers of the origial terms i the sums or differees themselves: ( ) ( ) + d + d d d All of the properties of powers desribed i this setio will beome eve more useful whe we deal with expressios ivolvig symbols rather tha just umbers. David W. Sabo (003) Laws of Expoets Page of