I can help break down the questions from the image and explain the solutions step by step.

1. First set of questions:

1a. $(a + 4b)^2$

This is a simple binomial expansion using the identity: $(a + 4b)^2 = a^2 + 2ab \cdot 4 + (4b)^2$ Thus, the expanded form is: $a^2 + 8ab + 16b^2$ The answer provided in the image appears to be "8", but that's unrelated to the expression itself, so it's possibly just part of feedback or grading.

1b. Simplify:

$\frac{\sqrt{50} + \sqrt{18}}{\sqrt{32 \times 2}}$ First, simplify the square roots: $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}, \quad \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}, \quad \sqrt{32 \times 2} = \sqrt{64} = 8$ Now, substitute into the expression: $\frac{5\sqrt{2} + 3\sqrt{2}}{8} = \frac{8\sqrt{2}}{8} = \sqrt{2}$ Thus, the simplified result is $\sqrt{2}$, and the answer provided is "5", possibly due to some grading or feedback.

2. Solve for $x$ in the equation:

$25^x + 25^x + 25^x + 25^x + 25^x = 5^{2021}$ The left side can be simplified as: $5 \times 25^x = 5 \times (5^2)^x = 5 \times 5^{2x} = 5^{1 + 2x}$ So, we now have: $5^{1 + 2x} = 5^{2021}$ Equating the exponents: $1 + 2x = 2021$ Solving for $x$: $2x = 2020 \quad \Rightarrow \quad x = 1010$ The answer provided in the image is "25", but the correct solution for $x$ is 1010.

3. Simplify the expression:

$\frac{(3x^2y^{-2}z^3)^3}{(x^{-2}y^2z^{-2})^2}$ First, simplify each part:

• For the numerator: $(3x^2y^{-2}z^3)^3 = 3^3x^{6}y^{-6}z^9 = 27x^6y^{-6}z^9$
• For the denominator: $(x^{-2}y^2z^{-2})^2 = x^{-4}y^4z^{-4}$ Now, divide the terms: $\frac{27x^6y^{-6}z^9}{x^{-4}y^4z^{-4}} = 27x^{6 - (-4)}y^{-6 - 4}z^{9 - (-4)} = 27x^{10}y^{-10}z^{13}$ Thus, the simplified expression is: $27x^{10}y^{-10}z^{13}$ The answer provided is "10", but the simplified form is $27x^{10}y^{-10}z^{13}$.

4. Simplify the logarithmic expression:

$2 \log 8 + 3 \log 9 + 5 \log 125$ We use the logarithmic identity: $a \log b = \log b^a$, so this becomes: $\log 8^2 + \log 9^3 + \log 125^5$ Simplify each term: $8^2 = 64, \quad 9^3 = 729, \quad 125^5 = 3125^2 = 9765625$ Now, use the property $\log a + \log b = \log (a \times b)$: $\log(64 \times 729 \times 9765625)$ This results in a very large number, but the expression can be simplified further through calculation.

Would you like more details or clarifications on any of these solutions? Here are 5 related questions you might find interesting:

1. How would you simplify other binomial expansions, such as $(a + b)^3$?
2. Can you solve logarithmic equations where the base is not 10, for example $\log_2 8$?
3. What is the general rule for solving exponential equations like $a^x = b$?
4. How do you apply logarithmic properties to simplify complex expressions?
5. Can you find the derivative of the expression $3x^2y^{-2}z^3$?

Tip: Always break down complex expressions into simpler parts, applying rules such as factoring or logarithmic identities step by step.