Question
14. The following terms form a pattern. \( \left(3^{3}\right)^{3},\left(3^{3}\right)^{4},\left(3^{3}\right)^{5},\left(3^{3}\right)^{6}, \ldots \) What would the fifth term be? Fully simplify your ans
Ask by Chang Pollard. in the United States
Dec 19,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
The fifth term is \(3^{21}\), which equals 10,460,353,203.
Solution
To determine the fifth term in the given pattern:
\[
\left(3^{3}\right)^{3}, \left(3^{3}\right)^{4}, \left(3^{3}\right)^{5}, \left(3^{3}\right)^{6}, \ldots
\]
**Step 1: Understand the Pattern**
Each term in the sequence can be expressed using exponent rules. Specifically, \(\left(a^b\right)^c = a^{b \times c}\).
**Step 2: Generalize the \(n\)-th Term**
The \(n\)-th term of the sequence is:
\[
\left(3^{3}\right)^{n+2} = 3^{3 \times (n+2)} = 3^{3n + 6}
\]
**Step 3: Find the Fifth Term**
For the fifth term, \(n = 5\):
\[
3^{3 \times 5 + 6} = 3^{15 + 6} = 3^{21}
\]
**Step 4: Simplify Completely**
Calculate \(3^{21}\):
\[
3^{21} = 10,\!460,\!353,\!203
\]
**Final Answer:**
The fifth term is \(10,\!460,\!353,\!203\).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find the fifth term in the pattern, we can follow the given structure. The general term can be expressed as \( \left(3^{3}\right)^{n} \), where \( n \) starts at 3 and increases by 1 for each subsequent term. So, for the fifth term, \( n = 3 + 4 = 7 \): \[ \left(3^{3}\right)^{7} = 3^{3 \cdot 7} = 3^{21}. \] Finally, \( 3^{21} \) is fully simplified, so the fifth term is \( 3^{21} \).
