Given the expression (2^3 * 3^2) / (2^2 * 3^4), let's solve this following each step given:
a. Apply the product rule:
When multiplying two powers that have the same base, you can add the exponents. However, in this case, we see that none of the powers in the numerator (i.e., top part of the fraction) or the denominator (i.e., bottom part of the fraction) have the same base and are being multiplied, hence this rule doesn't quite apply to this problem as it is.
b. Apply the quotient rule:
The quotient rule states that when dividing two powers that have the same base, you can subtract the exponents. In this case, we can apply this rule as we have bases 2 and 3 being divided.
For base 2: 2^(3-2) = 2^1
For base 3: 3^(2-4) = 3^-2
Our expression now looks like this: 2^1 / 3^-2
c. Simplify using the zero exponent rule:
The zero exponent rule states that any number (except 0) raised to the power of 0 is 1. In this problem, this rule does not apply as none of the exponents is 0.
d. Calculate the final result:
In this step, we calculate the final result using the powers we have determined for each base. When a number is raised to a negative exponent, it is actually equivalent to 1 divided by that number raised to the positive value of the exponent. So we could also write 3^-2 as 1/3^2. Now we just need to carry out the multiplications and divisions:
First, we consider base 2: 2^1 is just 2.
Next, we consider base 3: 1/3^2 is 1/9. Remember that 3^-2 is equivalent to 1/3^2.
Finally, we can calculate the result, which is 2 times 1/9 to give a final answer of approximately 0.2222222222222222.