Let's analyze and solve the problems from the uploaded image. The image contains two main problems, each with multiple sub-questions.

Problem 2: Simplify the following exponential expressions.

(a) $\frac{5^1}{5^2 \times 5^3}$
(b) $\frac{7^2 \times 6^4}{7^5}$
(c) $\left(\frac{1}{w}\right)^4 \times \left(\frac{1}{w}\right)$
(d) $\frac{3p^7}{p^3} \times 5p^1$

Solutions:

(a) $\frac{5^1}{5^2 \times 5^3}$

Simplify using the properties of exponents: $\frac{5^1}{5^5} = 5^{1-5} = 5^{-4} = \frac{1}{5^4}$

(b) $\frac{7^2 \times 6^4}{7^5}$

Simplify the expression involving $7$: $\frac{7^2}{7^5} \times 6^4 = 7^{2-5} \times 6^4 = 7^{-3} \times 6^4 = \frac{6^4}{7^3}$

(c) $\left(\frac{1}{w}\right)^4 \times \left(\frac{1}{w}\right)$

Simplify using the property of exponents: $\left(\frac{1}{w}\right)^{4+1} = \left(\frac{1}{w}\right)^5 = \frac{1}{w^5}$

(d) $\frac{3p^7}{p^3} \times 5p^1$

Simplify the expression involving $p$: $\frac{3p^7}{p^3} \times 5p^1 = 3p^{7-3} \times 5p^1 = 3p^4 \times 5p^1 = 15p^{4+1} = 15p^5$

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Problem 3: Simplify the following algebraic expressions.

(a) $\frac{-3p \sqrt{y}}{5by} \times \frac{-2b^3}{2x}$
(b) $\frac{7^5}{a^4} \times \left(\frac{15}{16}\right)^{3/2}$
(c) $\frac{3^5}{2^4 \times 5^2}$
(d) $\frac{2^{n+2} \times 2^7}{2^{n+5}}$

Solutions:

(a) $\frac{-3p \sqrt{y}}{5by} \times \frac{-2b^3}{2x}$

First, simplify the multiplication of the fractions: $\frac{(-3p \sqrt{y}) \times (-2b^3)}{(5by) \times (2x)} = \frac{6pb^3 \sqrt{y}}{10bxy} = \frac{3pb^2 \sqrt{y}}{5xy}$

(b) $\frac{7^5}{a^4} \times \left(\frac{15}{16}\right)^{3/2}$

Simplify the multiplication: $\frac{7^5}{a^4} \times \left(\frac{15}{16}\right)^{3/2}$ Here, the expression is already simplified as much as possible unless further specific simplifications for $\left(\frac{15}{16}\right)^{3/2}$ are requested.

(c) $\frac{3^5}{2^4 \times 5^2}$

Calculate the powers of the numbers: $\frac{243}{16 \times 25} = \frac{243}{400}$

(d) $\frac{2^{n+2} \times 2^7}{2^{n+5}}$

Simplify using the properties of exponents: $\frac{2^{n+2+7}}{2^{n+5}} = \frac{2^{n+9}}{2^{n+5}} = 2^{n+9-n-5} = 2^{4}$

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Would you like any details on any of these solutions, or do you have any further questions?

Related Questions:

1. What are the properties of exponents used in these simplifications?
2. How do you simplify algebraic expressions involving square roots?
3. What steps should you take when simplifying fractions involving polynomials?
4. How can the power of a fraction be simplified when raised to another power?
5. What are some common mistakes to avoid when simplifying expressions with multiple variables?

Tip:

Always remember to apply the properties of exponents consistently, especially when dealing with negative exponents or when combining powers.