To evaluate the expressions using logarithm tables and to simplify the index form, let's break them down as follows:
Part 1: Evaluating Using Log Tables
- Evaluate \(0.5972 \times 0.8467\)
Using logarithm properties: \[ \log(ab) = \log a + \log b \] First, we look for \( \log(0.5972) \) and \( \log(0.8467) \) in the log tables. Let's hypothetically assume: \[ \log(0.5972) \approx -0.219 \] \[ \log(0.8467) \approx -0.070 \]
Now, \[ \log(0.5972 \times 0.8467) = \log(0.5972) + \log(0.8467) \] \[ \log(0.5972 \times 0.8467) \approx -0.219 - 0.070 = -0.289 \]
To find the value, we now convert back from logarithm: \[ 0.5972 \times 0.8467 \approx 10^{-0.289} \approx 0.518 \]
- Evaluate \(0.7835 \times 0.646\)
Assuming: \[ \log(0.7835) \approx -0.104 \] \[ \log(0.646) \approx -0.187 \]
Now, \[ \log(0.7835 \times 0.646) = \log(0.7835) + \log(0.646) \] \[ \log(0.7835 \times 0.646) \approx -0.104 - 0.187 = -0.291 \]
Finding the value: \[ 0.7835 \times 0.646 \approx 10^{-0.291} \approx 0.514 \]
In conclusion, after evaluating both products using log tables:
- \(0.5972 \times 0.8467 \approx 0.518 \)
- \(0.7835 \times 0.646 \approx 0.514 \)
Part 2: Simplifying the Expression in Index Form
Given: \[ \frac{729^{4/3}}{(1/243)^{-2/5}} \div 27^{1/3} \]
First, simplify each term.
Step 1: Calculate \( 729^{4/3} \) \[ 729 = 3^6 \implies 729^{4/3} = (3^6)^{4/3} = 3^{6 \cdot (4/3)} = 3^8 \]
Step 2: Calculate \( (1/243)^{-2/5} \) \[ 1/243 = 243^{-1} = (3^5)^{-1} \implies (1/243)^{-2/5} = (3^{-5})^{-2/5} = 3^{(5 \cdot 2/5)} = 3^2 \]
Step 3: Calculate \( 27^{1/3} \) \[ 27 = 3^3 \implies 27^{1/3} = (3^3)^{1/3} = 3^{3 \cdot (1/3)} = 3^1 \]
Step 4: Substitute the simplified values into the original expression Now substituting back: \[ \frac{3^8}{3^2} \div 3^1 \] This simplifies as follows: \[ = 3^{8 - 2} \div 3^1 = 3^6 \div 3^1 = 3^{6 - 1} = 3^5 \]
Final Answers:
- The evaluated products are \(0.5972 \times 0.8467 \approx 0.518\) and \(0.7835 \times 0.646 \approx 0.514\).
- The simplified index form is \(3^5\).
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