Which of the following expressions is/are equivalent to 18 x 6^(−2x+1)? There may be more than one correct answer. Select all that apply.
a) 108^x/36
b) 3 x 36/36^x
c) 3 x 36^(1-x)
d) 108/36^x
e)3^x
To determine which expressions are equivalent to 18 x 6^(−2x+1), we can simplify the original expression and then compare it to each of the given answer choices.
First, let's simplify the original expression:
18 x 6^(-2x+1)
To simplify the expression, we can rewrite 6^(-2x+1) as 1/(6^(2x-1)). Now we have:
18 x 1/(6^(2x-1))
To make the comparisons easier, let's rewrite the given answer choices in the same format:
a) 108^x/36 = (12^2)^x / 6^2 = 12^2x / 6^2
b) 3 x 36/36^x = 3 x 6^2 / (6^2)^x = 3 x 6^2 / 6^2x
c) 3 x 36^(1-x) = 3 x 6^2(1-x) = 3 x 6^2 -2x
d) 108/36^x = 12^2 / (6^2)^x = 12^2 / 6^2x
e) 3^x = 3^x
Now, let's compare each answer choice to the simplified expression:
a) 12^2x / 6^2 - Not equivalent, the exponents do not match.
b) 3 x 6^2 / 6^2x - Not equivalent, the exponents do not match.
c) 3 x 6^2 -2x - Not equivalent, the exponents do not match.
d) 12^2 / 6^2x - Equivalent, the exponents match.
e) 3^x - Not equivalent, the exponents do not match.
So, the answer choices that are equivalent to 18 x 6^(−2x+1) are:
d) 108/36^x
Therefore, option d is the correct answer.