Question

Use log tables to evaluate 4mks
0.5972 × 0.8467
0.7835 × 0.646
2. Simplify the following, leaving the answer in simplified index form. 3mks
7294/3
× (1/243)-2/5
÷ 271/3

Answer 1

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

1. **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:
\[
0.5972 \times 0.8467 \approx 10^{-0.289} \approx 0.518
\]

2. **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:
1. The evaluated products are \(0.5972 \times 0.8467 \approx 0.518\) and \(0.7835 \times 0.646 \approx 0.514\).
2. The simplified index form is \(3^5\).