Let me transcribe the content of the image and solve the problems for you.

Image Content

1. Expression 1:
   $\log_2 8 + \log_2 16 - 4 \log_2 2$

2. Expression 2:
   $\log_4 16$

3. Expression 3: Without using tables, evaluate:
   (i) $(0.64)^{1/2}$
   (ii) $\left(\frac{25}{9}\right)^{3/2}$
   (iii) $(0.36)^{1/2}$
   (iv) $\left(\frac{125}{27}\right)^{-2/3}$

Solution

1. Solving $\log_2 8 + \log_2 16 - 4\log_2 2$:

• Recall the property: $\log_b(x \cdot y) = \log_b x + \log_b y$, and $\log_b(x^n) = n \cdot \log_b x$.
• Breakdown:
  $\log_2 8 = \log_2(2^3) = 3, \quad \log_2 16 = \log_2(2^4) = 4, \quad \text{and } 4 \log_2 2 = 4 \cdot 1 = 4.$
• Substitute:
  $\log_2 8 + \log_2 16 - 4 \log_2 2 = 3 + 4 - 4 = 3.$
• Answer: $3$.

2. Solving $\log_4 16$:

• Convert to base 4:
  $\log_4 16 = \log_4(4^2) = 2 \cdot \log_4 4 = 2 \cdot 1 = 2.$
• Answer: $2$.

3. Evaluations Without Using Tables

(i) $(0.64)^{1/2}$:
$(0.64)^{1/2} = \sqrt{0.64} = 0.8.$
Answer: $0.8$.

(ii) $\left(\frac{25}{9}\right)^{3/2}$:

• Apply the exponent rule:
  $\left(\frac{25}{9}\right)^{3/2} = \left(\sqrt{\frac{25}{9}}\right)^3 = \left(\frac{\sqrt{25}}{\sqrt{9}}\right)^3 = \left(\frac{5}{3}\right)^3 = \frac{125}{27}.$
• Answer: $\frac{125}{27}$.

(iii) $(0.36)^{1/2}$:
$(0.36)^{1/2} = \sqrt{0.36} = 0.6.$
Answer: $0.6$.

(iv) $\left(\frac{125}{27}\right)^{-2/3}$:

• Apply the exponent rule:
  [ \left(\frac{125}{27}\right)^{-2/3} = \frac{1}{\left(\frac{125}{27}\right)^{2/3}} = \frac{1}{\left(\frac{\sqrt[3]{125}}{\sqrt[3]{27}}\right)^2}. ]
• Simplify cube roots:
  [ \sqrt[3]{125} = 5, \quad \sqrt[3]{27} = 3 \quad \implies \left(\frac{\sqrt[3]{125}}{\sqrt[3]{27}}\right)^2 = \left(\frac{5}{3}\right)^2 = \frac{25}{9}. ]
• Reciprocal:
  $\left(\frac{125}{27}\right)^{-2/3} = \frac{1}{\frac{25}{9}} = \frac{9}{25}.$
• Answer: $\frac{9}{25}$.

Summary of Answers:

1. $3$
2. $2$
3. (i) $0.8$, (ii) $\frac{125}{27}$, (iii) $0.6$, (iv) $\frac{9}{25}$.

Would you like more details or have any specific questions? Here are 5 related questions to explore:

1. What are other properties of logarithms that are useful for simplifications?
2. How do fractional exponents relate to roots in mathematics?
3. How can negative exponents be interpreted with fractions?
4. How do we approach solving logarithms with bases other than 10 or 2?
5. Can you explain why cube roots are used in $\frac{125}{27}$ specifically?

Tip: When working with fractional exponents, remember $x^{m/n}$ means the $n$-th root of $x^m$.