Let's solve the two parts of this question step by step.

Problem 1a

Evaluate: $5\sqrt{28} - 3\sqrt{63} + 5\sqrt{112}$

1. Simplify each square root:

   • $\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$
   • $\sqrt{63} = \sqrt{9 \times 7} = 3\sqrt{7}$
   • $\sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7}$

2. Rewrite the expression using these simplifications: $5\sqrt{28} - 3\sqrt{63} + 5\sqrt{112} = 5 \times 2\sqrt{7} - 3 \times 3\sqrt{7} + 5 \times 4\sqrt{7}$

3. Calculate each term:

   • $5 \times 2\sqrt{7} = 10\sqrt{7}$
   • $-3 \times 3\sqrt{7} = -9\sqrt{7}$
   • $5 \times 4\sqrt{7} = 20\sqrt{7}$

4. Combine the terms: $10\sqrt{7} - 9\sqrt{7} + 20\sqrt{7} = (10 - 9 + 20)\sqrt{7} = 21\sqrt{7}$

So, the answer to part (a) is: $21\sqrt{7}$

Problem 1b

Evaluate: $\frac{1}{5} \log 16 + \frac{1}{2} \log 25 - \frac{1}{3} \log \frac{27}{81}$

1. Simplify each term using the power rule of logarithms, $a \log b = \log b^a$:

   • $\frac{1}{5} \log 16 = \log 16^{\frac{1}{5}} = \log 2$
   • $\frac{1}{2} \log 25 = \log 25^{\frac{1}{2}} = \log 5$
   • $\frac{1}{3} \log \frac{27}{81} = \log \left(\frac{27}{81}\right)^{\frac{1}{3}} = \log \frac{3}{9} = \log \frac{1}{3}$

2. Combine the logs: $\log 2 + \log 5 - \log \frac{1}{3}$

3. Use the properties of logarithms to simplify further: $\log (2 \times 5) + \log 3 = \log (10 \times 3) = \log 30$

So, the answer to part (b) is: $\log 30$

Summary

• Answer to 1a: $21\sqrt{7}$
• Answer to 1b: $\log 30$

Would you like more details on any of these steps, or have further questions?

Further Questions

1. How are logarithmic properties applied to simplify expressions?
2. What are the steps to simplify square roots in expressions?
3. How can we combine terms with different square root factors?
4. How do fractional exponents relate to roots in logarithmic simplifications?
5. What are some additional examples of simplifying complex logarithmic expressions?

Tip: Always break down expressions step-by-step to simplify radicals and logs. This approach minimizes errors and clarifies each part of the process.