Weston Lowe
01/02/2023 · Senior High School
(3) \( \frac{(81)^{1 / 4}}{16} \times 2^{-1} \) (3) \( \left.(216)^{13}\right)^{-2} \)
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Tutor-Verified Answer
Step-by-step Solution
Let's simplify each expression step by step.
### Expression 1: \( \frac{(81)^{1 / 4}}{16} \times 2^{-1} \)
1. **Calculate \( (81)^{1/4} \)**:
\[
81 = 3^4 \quad \text{so} \quad (81)^{1/4} = (3^4)^{1/4} = 3^{4 \cdot \frac{1}{4}} = 3^1 = 3
\]
2. **Substitute back into the expression**:
\[
\frac{(81)^{1 / 4}}{16} \times 2^{-1} = \frac{3}{16} \times 2^{-1}
\]
3. **Calculate \( 2^{-1} \)**:
\[
2^{-1} = \frac{1}{2}
\]
4. **Combine the fractions**:
\[
\frac{3}{16} \times \frac{1}{2} = \frac{3 \cdot 1}{16 \cdot 2} = \frac{3}{32}
\]
So, the simplified result of the first expression is:
\[
\frac{3}{32}
\]
### Expression 2: \( (216)^{13})^{-2} \)
1. **Apply the exponent rule**:
\[
(a^m)^n = a^{m \cdot n}
\]
Thus,
\[
(216)^{13})^{-2} = (216)^{13 \cdot (-2)} = (216)^{-26}
\]
2. **Calculate \( (216)^{-26} \)**:
\[
(216)^{-26} = \frac{1}{(216)^{26}}
\]
3. **Express \( 216 \) in terms of its prime factors**:
\[
216 = 6^3 = (2 \cdot 3)^3 = 2^3 \cdot 3^3
\]
Therefore,
\[
(216)^{26} = (2^3 \cdot 3^3)^{26} = 2^{3 \cdot 26} \cdot 3^{3 \cdot 26} = 2^{78} \cdot 3^{78}
\]
4. **Substituting back**:
\[
(216)^{-26} = \frac{1}{2^{78} \cdot 3^{78}} = 2^{-78} \cdot 3^{-78}
\]
So, the simplified result of the second expression is:
\[
2^{-78} \cdot 3^{-78}
\]
### Final Results:
1. \( \frac{3}{32} \)
2. \( 2^{-78} \cdot 3^{-78} \)
Quick Answer
1. \( \frac{3}{32} \)
2. \( 2^{-78} \cdot 3^{-78} \)
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