Exponents!

[cookies]

Okay, I have a few questions about putting exponents in simplest form ..

 

Example

 

5^5 x 5^5 = 5^10 just add the exponents

 

 

9^5 x 8^5 = 72^10??

 

 

I understand how to do it if its the same number but i cant understand if its diff please tell me..

[imp]

Okay, I have a few questions about putting exponents in simplest form ..

 

Example

 

5^5 x 5^5 = 5^10 just add the exponents

 

 

9^5 x 8^5 = 72^10??

 

 

I understand how to do it if its the same number but i cant understand if its diff please tell me..

 

Cookies, the number being raised to a power is called the "base". To multiply two or more bases raised to some power, we add the exponents, keeping the base the same. Two non-identical bases raised to some power CANNOT have their exponents added. FYI, similarly, two like bases raised to some power may be DIVIDED by subtracting their exponents. Imp.

[cookies]

So how would i do the 1 with diff base?

[Bignose]

Well, firstly, you should have some software that can compute these numbers for you. Both Excel and Windows calculator can calculate these numbers ... 9^5 + 8^5 = 1934927632 whereas 72^10 = 3743906242624487424, clearly not the same. However, 72^5 = 1934927632. If and only if the exponents are the same, you can mulitply the bases: 9^5 * 8^5 = (9*8)^5 = 72^5. But, if it is 9^4 + 8^5, there is no simplification possible.

[cookies]

thanks so much!

[Tom Mattson]

So how would i do the 1 with diff base?

 

You can do it like this.

 

[math]a^mb^m=(a \cdot a \cdot\cdot\cdot a)(b \cdot b \cdot\cdot\cdot b)[/math],

 

where each of the parentheses contains m factors. You can then use the commutative and associative properties of multiplication to rearrange the various factors of a and b as follows.

 

[math](ab)\cdot(ab)\cdot\cdot\cdot(ab)[/math]

 

Now you have m factors of (ab). But this is precisely (ab)^m.

 

So we arrive at the rule: [math]a^mb^m=(ab)^m[/math]

 

So back to your example...

 

[math]9^5 \cdot 8^5=(9 \cdot 8)^5=72^5[/math]

[the tree]

Using Tom's method, you can also note that a^mb^(m+p)=(ab)^mb^p and probably lots of other noteworthy identities.