Final answer:
To find the quotient (8^3x+1) ÷ (2^x), we rewrite 8 as 2^3 and apply the properties of exponents to divide the coefficients and subtract the exponents, resulting in the final answer of 2^{8x+3}.
Step-by-step explanation:
Division of Exponentials Example
To find the quotient (8^3x+1) ÷ (2^x), we need to use the properties of exponents, specifically the division of exponentials. According to the rules, when dealing with exponential terms, we divide the digit term of the numerator by the digit term of the denominator and subtract the exponents of the exponential terms. Since 8 is equal to 2 cubed (2^3), we can rewrite the problem as (2^3)^{3x+1} ÷ (2^x). By using the rule of exponents that states (a^b)^c = a^(bc), we get 2^{3(3x+1)} ÷ 2^x.
Now, we subtract the exponent in the denominator (x) from the exponent in the numerator (9x+3) according to the division rule. This leads us to 2^{9x+3-x}, which simplifies to 2^{8x+3}. This is our final quotient in terms of the original variables provided in the question.