Posted by dj on Sunday, August 16, 2009 at 4:29pm.
Simplify each of the following, if possible. Write your answers in exponential form, a^b
3^4*3^7*3^6
2^80+3*2^80

math?  MathMate, Sunday, August 16, 2009 at 4:40pm
There are two different cases here.
3^4*3^7*3^6 is a pure multiplicative expression, so the rules of priority and rules of exponents are applicable.
1. exponentiation has priority over multiplication
2. multiplication of two numbers to the same base a raised to exponents x and y yields a result of a^{x+y}.
So
3^4*3^7*3^6
= (3^{4})*(3^{7})*(3^{6})
= 3^{4+7+6}
=3^{17}
The second expression, 2^80+3*2^80, has two terms with a common factor 2^{80}. So the simplification consists in factorization of the expression, namely
2^80+3*2^80
= 1*2^{80} + 3*2^{80}
= (1+3)2^{80}
= 4*2^{80} 
math?  dj, Sunday, August 16, 2009 at 4:45pm
In the second problem why do I put a 1 before 2^80?

math?  bobpursley, Sunday, August 16, 2009 at 4:58pm
you don't have to, but on the second step it makes factorization more apparent.

math?  Anonymous, Sunday, August 16, 2009 at 5:04pm
thats why the answer not 3^2^280 I am not understanding the answer

math?  dj, Sunday, August 16, 2009 at 5:06pm
thats why the answer not 3^2^280 I am not understanding the answer
you don't multiply 3*2=6^80 
math?  MathMate, Sunday, August 16, 2009 at 5:39pm
The rules of priority of operations requires us to do exponents first, so
3*2^80 is the same as 3*(2^80). That is to say, we do the exponentiation of 2 to the power of 80 before multiplication.
If we multiply before exponentiation, we are not following the rules of priority of operations.
Then come the rules of exponentiation, some of the basic ones are :
a^{0} = 1 for any value of a≠0
a^{1} = a
a^{2} = a*a
a^{3} = a*a*a, etc.
a^{1} = 1/a
a^{2} = 1/(a*a)
a^{3} = 1/(a*a*a), etc.
a^{x} * a^{y} = a^{x+y}
This rule can be applied only if the two bases (a) are the same.
This also means that we cannot simplify a^{x}*b^{y} for general values of a and b.
Since the expression
3*2^{80}
=3^{1} * 2^{80}
has 3 and 2 as bases for the exponents, we cannot simplify the expression by the rules of exponents.