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
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 ax+y.
So
3^4*3^7*3^6
= (34)*(37)*(36)
= 34+7+6
=317
The second expression, 2^80+3*2^80, has two terms with a common factor 280. So the simplification consists in factorization of the expression, namely
2^80+3*2^80
= 1*280 + 3*280
= (1+3)280
= 4*280
In the second problem why do I put a 1 before 2^80?
you don't have to, but on the second step it makes factorization more apparent.
thats why the answer not 3^2^280 I am not understanding the answer
thats why the answer not 3^2^280 I am not understanding the answer
you don't multiply 3*2=6^80
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 :
a0 = 1 for any value of a≠0
a1 = a
a2 = a*a
a3 = a*a*a, etc.
a-1 = 1/a
a-2 = 1/(a*a)
a-3 = 1/(a*a*a), etc.
ax * ay = ax+y
This rule can be applied only if the two bases (a) are the same.
This also means that we cannot simplify ax*by for general values of a and b.
Since the expression
3*280
=31 * 280
has 3 and 2 as bases for the exponents, we cannot simplify the expression by the rules of exponents.
To simplify each of the given expressions, we need to combine the bases and add the exponents. Let's solve them step by step.
1. Simplifying the expression 3^4 * 3^7 * 3^6:
To simplify, we combine the bases (which are all 3) and add the exponents. So, we add 4, 7, and 6 together:
3^(4+7+6) = 3^17
Therefore, the expression simplifies to 3^17.
2. Simplifying the expression 2^80 + 3 * 2^80:
Here, we have the same base (2) with exponents that are added to each other. We can combine them by factoring out 2^80 from both terms:
2^80(1 + 3)
Simplifying further:
2^80(4)
Therefore, the expression simplifies to 4 * 2^80.
Hence, the simplified forms of the given expressions in exponential form are:
1. 3^4 * 3^7 * 3^6 = 3^17
2. 2^80 + 3 * 2^80 = 4 * 2^80