Let's go through the questions shown on the board, which are related to the properties of exponents:

L1: Evaluate

1. a) $20^0$

   • Any non-zero number raised to the power of 0 is 1.
   • $20^0 = 1$

2. b) $5^3 \times 5^4$

   • Use the property $a^m \times a^n = a^{m+n}$.
   • $5^3 \times 5^4 = 5^{3+4} = 5^7$

3. c) $\frac{5^8}{5^7}$

   • Use the property $\frac{a^m}{a^n} = a^{m-n}$.
   • $\frac{5^8}{5^7} = 5^{8-7} = 5^1 = 5$

4. d) $(5^2)^3$

   • Use the property $(a^m)^n = a^{m \times n}$.
   • $(5^2)^3 = 5^{2 \times 3} = 5^6$

L2: Evaluate

1. a) $5^{-4}$

   • Use the property $a^{-n} = \frac{1}{a^n}$.
   • $5^{-4} = \frac{1}{5^4}$

2. b) $\frac{1}{3^{-5}}$

   • Simplify the negative exponent using $\frac{1}{a^{-n}} = a^n$.
   • $\frac{1}{3^{-5}} = 3^5$

3. c) $\frac{4^{-8}}{4^{-5}}$

   • Use $\frac{a^m}{a^n} = a^{m-n}$.
   • $\frac{4^{-8}}{4^{-5}} = 4^{-8 - (-5)} = 4^{-3} = \frac{1}{4^3}$

4. d) $(4^{-2})^{-5}$

   • Use $(a^m)^n = a^{m \times n}$.
   • $(4^{-2})^{-5} = 4^{(-2) \times (-5)} = 4^{10}$

Extension:

Evaluate $\frac{3^{-9}}{3^{-6}} \times (3^2)^5$

1. Simplify $\frac{3^{-9}}{3^{-6}}$ using $\frac{a^m}{a^n} = a^{m-n}$:

   • $\frac{3^{-9}}{3^{-6}} = 3^{-9 - (-6)} = 3^{-3}$

2. Simplify $(3^2)^5$ using $(a^m)^n = a^{m \times n}$:

   • $(3^2)^5 = 3^{2 \times 5} = 3^{10}$

3. Multiply the results:

   • $3^{-3} \times 3^{10} = 3^{-3 + 10} = 3^7$

Would you like further details or have any questions?

5 Related Questions:

1. What is the general rule for multiplying powers with the same base?
2. How do you handle negative exponents in fractional form?
3. How does raising a power to another power work?
4. What is the rule for dividing powers with the same base?
5. How do you simplify an expression with a zero exponent?

Tip:

When dealing with exponents, always remember to apply the correct exponent rule based on the operation (multiplication, division, or power to power).