I can help you solve the problems visible in the image. Let's start with identifying and solving the ones that can be clearly seen.

Problems:

1. 5. $(3x^4y) - (-3x^{-2}y^6)$:

   • This is a subtraction problem involving powers of $x$ and $y$.

2. 6. $\left( \frac{21x^6 y^5}{3x^7 y^3} \right)^0$:

   • This is an expression raised to the power of 0, which simplifies directly.

3. 7. $f(x) = -2^x + 1$:

   • The function asks you to evaluate it for different values of $x$.

Let's start with these. I will solve each problem step by step:

Problem 5:

$(3x^4y) - (-3x^{-2}y^6)$

• First, remove the negative sign: $3x^4y + 3x^{-2}y^6$
• These two terms are not like terms, so this is the simplified form: $3x^4y + 3x^{-2}y^6$

Problem 6:

$\left( \frac{21x^6 y^5}{3x^7 y^3} \right)^0$

• Simplify the expression inside the parentheses first: $\frac{21x^6 y^5}{3x^7 y^3} = 7x^{-1}y^2$
• Since the entire expression is raised to the power of 0: $(7x^{-1}y^2)^0 = 1$
• Any non-zero expression raised to the power of 0 is 1.

Problem 7:

Evaluate $f(x) = -2^x + 1$ for different values of $x$. For example:

• If $x = 1$: $f(1) = -2^1 + 1 = -2 + 1 = -1$
• If $x = 2$: $f(2) = -2^2 + 1 = -4 + 1 = -3$
• If $x = 0$: $f(0) = -2^0 + 1 = -1 + 1 = 0$

Let me know if you'd like to go over any of these in more detail, or if you'd like help with other problems!

Follow-up Questions:

1. What is the general rule for simplifying terms with exponents?
2. How do you handle subtraction when negative terms are involved in algebraic expressions?
3. What are the properties of any term raised to the power of zero?
4. How does the function $f(x) = -2^x + 1$ behave for negative values of $x$?
5. How would you simplify a similar expression with higher exponents, like $\left( \frac{50x^{10}}{10x^5} \right)$?

Tip:

Always simplify the terms inside parentheses first, especially if the expression involves exponents, before applying any additional operations like raising it to a power.