Let’s break down the math problems in the image step-by-step.

1. Problems Overview:

The image contains 9 distinct math problems. They cover:

• Exponents and operations with powers
• Simplifications
• Solving for $x$
• Working with fractions and exponents

Below is a solution outline for each problem:

Problem 1

Simplify: $\left( \frac{3}{2} \right)^2 + \left( \frac{5}{3} \right)^2 + \left( \frac{4}{3} \right)^2$

• Square each term and sum the results.

Problem 2

Simplify the product: $(3^x)^2 \cdot (3^x)^3 \cdot (3^x)^4$

• Use the exponent multiplication rule: $a^m \cdot a^n = a^{m+n}$.

Problem 3

Simplify the expression: $\left( \frac{3}{2} \right)^1 \cdot \left( \frac{1}{2} \right)^1 \cdot x^1 = 6x^2$

• Solve for $x$.

Problem 4 (a)

Solve for $x$ in: $5^x + 5^{-x} = 5^2$

• Use substitutions to simplify.

Problem 4 (b)

Solve for $x$ in: $\frac{2x - 2}{3x - 5} = \frac{1}{9}$

• Cross-multiply and solve for $x$.

Problem 4 (c)

Simplify: $\left( \frac{3}{2} \right)^{x+1} - \left( \frac{3}{2} \right)^x = \left( \frac{3}{2} \right)^1$

• Factor out common powers.

Problem 4 (d)

Simplify and solve for $x$: $2 \cdot \left( \frac{1}{2} \right)^{x+1} = \left( \frac{1}{2} \right)^x$

• Apply exponent rules.

Problem 5

Determine the number $k$ such that: $\left( \frac{3}{4} \right)^k \cdot N = \left( \frac{3}{4} \right)^7$

• Solve for $N$.

Problem 6

Simplify: $\frac{2 \cdot 10^{-2} + 1.25}{5 \cdot 10^{-2}}$

• Combine terms in the numerator and divide.

Would you like me to solve any specific problem in detail or all of them step-by-step? Let me know which part you want me to focus on!

5 Related Questions

1. How do you simplify expressions with powers and roots?
2. How do you solve exponential equations like $a^x + a^{-x} = k$?
3. What are the rules for solving rational equations?
4. How do you factor terms with common powers?
5. How do you simplify complex fractions with exponents?

Tip: When dealing with powers, always group terms with the same base and apply exponent rules carefully!