Which of the following expressions is/are equivalent to 1/12 x 36^t? There may be more than one correct answer. Select all that apply.
a) 1/2 x 6 x 36^t
b) 1/2 x 6^-1 x 36^t
c) 1/2 x 6^2t-1
d) (2^-1)^2 x 3^2t-1
e) 2^2t-1 x 3^2t-1
To determine the equivalent expressions to 1/12 x 36^t, let's break it down step by step.
1. Start with the expression: 1/12 x 36^t
2. Rewrite 36 as 6^2 since 6^2 = 36. This gives us: 1/12 x (6^2)^t
3. Use the exponent rule that (a^b)^c = a^(b*c). Applying this, we get: 1/12 x 6^(2t)
Now, let's examine each option and simplify them to see which ones are equivalent:
a) 1/2 x 6 x 36^t
⇒ Simplify: 1/2 x 6^1 x 36^t
⇒ Simplify: 1/2 x 6^(2t)
This is equivalent to the original expression.
b) 1/2 x 6^-1 x 36^t
⇒ Simplify: 1/2 x (6^(-1)) x 36^t
⇒ Simplify: 1/2 x 6^(-1) x 36^t
⇒ Simplify: 1/2 x (1/6) x 36^t
⇒ Simplify: 1/2 x (1/6) x 6^(2t)
This is not equivalent to the original expression.
c) 1/2 x 6^2t-1
⇒ Since the exponent is outside the parentheses, apply the power rule by multiplying the exponents:
1/2 x 6^(2t) x 6^(-1)
⇒ Simplify: 1/2 x 6^(2t) / 6
This is not equivalent to the original expression.
d) (2^-1)^2 x 3^2t-1
⇒ Simplify: 1/2^2 x 3^(2t-1)
⇒ Simplify: 1/4 x 3^(2t-1)
This is not equivalent to the original expression.
e) 2^2t-1 x 3^2t-1
This expression cannot be simplified further.
From the analysis, the equivalent expressions to 1/12 x 36^t are a) 1/2 x 6 x 36^t.
Therefore, the correct answer is a).