It looks like you've uploaded an image with various expressions involving exponents. Let's go through each problem step by step and evaluate the expressions:

1. $3^0 + 5^0$

• Any non-zero number raised to the power of 0 equals 1.
• $3^0 = 1$ and $5^0 = 1$
• Therefore, $1 + 1 = 2$

2. $2^4 \times 2^{-2}$

• Using the rule of exponents $a^m \times a^n = a^{m+n}$:
• $2^4 \times 2^{-2} = 2^{4-2} = 2^2 = 4$

3. $3^4 + 3^3$

• $3^4 = 81$ and $3^3 = 27$
• So, $81 + 27 = 108$

4. $2^0 \times 4^{-2}$

• $2^0 = 1$, and $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$
• Therefore, $1 \times \frac{1}{16} = \frac{1}{16}$

5. $2^2 + 2^2$

• $2^2 = 4$, so $4 + 4 = 8$

6. $2^4 \times 2^0 + 2^3$

• Using the same exponent rule $a^m \times a^n = a^{m+n}$:
• $2^4 \times 2^0 = 2^{4+0} = 2^4 = 16$
• So, $16 + 2^3 = 16 + 8 = 24$

7. $(6 + 5)^0 - 2^3$

• $(6 + 5)^0 = 1$ because anything raised to the power of 0 is 1.
• $2^3 = 8$
• Therefore, $1 - 8 = -7$

8. $\left(\frac{2}{3}\right)^{-1} + 4^{-1}$

• $\left(\frac{2}{3}\right)^{-1} = \frac{3}{2}$ because the inverse of a fraction $\frac{a}{b}$ is $\frac{b}{a}$.
• $4^{-1} = \frac{1}{4}$
• So, $\frac{3}{2} + \frac{1}{4}$
  • To add these fractions, find a common denominator: $\frac{3}{2} = \frac{6}{4}$, so
  • $\frac{6}{4} + \frac{1}{4} = \frac{7}{4}$

Let me know if you'd like further details on any of the solutions or if you have any additional questions!

Related questions:

1. What happens when you multiply exponents with the same base?
2. How do you deal with negative exponents in fractions?
3. Can zero raised to any number result in a nonzero value?
4. How do you simplify expressions involving both positive and negative exponents?
5. What is the difference between multiplication and division rules for exponents?

Tip: When simplifying expressions with exponents, always look for opportunities to combine like terms using the rules of exponents!