Question

which of these expressions are not equivalent to the expression $1.5^{3t + 2}$
$\circ$ $2.25(1.5)^{3t}$
$\circ$ $2.25(3.375)^{t}$
$\circ$ $(1.5^{3t}) + (1.5)^{2}$
$\circ$ $(1.5^{2})(1.5)^{3t}$

Explanation:

Step1: Recall exponent rule \(a^{m+n}=a^m\cdot a^n\)

For \(1.5^{3t + 2}\), by the rule \(a^{m + n}=a^m\cdot a^n\), we have \(1.5^{3t+2}=1.5^{3t}\cdot1.5^{2}\). Calculate \(1.5^{2}=2.25\), so \(1.5^{3t+2}=2.25\cdot1.5^{3t}\), so the first and fourth options are equivalent.

Step2: Simplify \(1.5^{3t}\)

We know that \(1.5^{3}=1.5\times1.5\times1.5 = 3.375\), so \(1.5^{3t}=(1.5^{3})^{t}=3.375^{t}\). Then \(1.5^{3t + 2}=2.25\times3.375^{t}\), so the second option is equivalent.

Step3: Analyze the third option

The third option is \((1.5^{3t})+(1.5)^{2}\), which is a sum of two terms, while \(1.5^{3t+2}\) is a product of two terms (\(1.5^{3t}\) and \(1.5^{2}\)). Since \(a^{m + n}
eq a^m + a^n\) in general (exponent rule is for multiplication, not addition), the third option is not equivalent.

Answer:

\((1.5^{3t})+(1.5)^{2}\) (the third option)