Let's go through each problem step by step.

1. Find $x$ in $2(4^x) - 10(2^x) + 8 = 0$

• Rewrite $4^x$ as $(2^2)^x = (2^x)^2$.
• Let $y = 2^x$, so $4^x = y^2$.
• Substitute into the equation: $2y^2 - 10y + 8 = 0$.
• Simplify and solve this quadratic equation for $y$.
• Once $y$ is found, solve for $x$ by setting $y = 2^x$.

2. Solve for $x$ in $2^{2x} - 18 \cdot 2^x = 40$

• Let $y = 2^x$, then $2^{2x} = y^2$.
• Substitute into the equation: $y^2 - 18y - 40 = 0$.
• Solve the resulting quadratic equation for $y$.
• Once $y$ is found, solve for $x$ by setting $y = 2^x$.

3. Solve for $x$ given $\left(\frac{1}{8}\right)^x \cdot 64 = 25\%$

• Rewrite $64$ as $8^2$ and $25\%$ as $\frac{1}{4}$.
• The equation becomes $\left(\frac{1}{8}\right)^x \cdot 8^2 = \frac{1}{4}$.
• Simplify and solve for $x$.

4. Simplify $\frac{3n^3 - 3 \times 3^n}{4 \times 3^{n+2}}$

• Rewrite and simplify the terms in the numerator and the denominator.
• Factor and reduce the expression.

5. Solve for $x$ in the equation $64^x - 121 = 17 - 4 \cdot 3^x$

• Rewrite $64$ as $4^3$ or $2^6$ depending on the context.
• Rearrange and simplify the terms.
• Solve for $x$ by isolating it on one side.

6. Find $x$ if $3^{2x+3} + 1 = 28$

• Start by isolating $3^{2x+3}$.
• Solve for $x$ by taking the logarithm on both sides after simplifying.

Would you like me to go into detail for any specific problem or provide a complete solution for each?

Follow-up Questions

1. How do we solve equations involving exponential functions systematically?
2. What is the best way to simplify complex fractions involving exponents?
3. How can substitution help simplify exponential equations?
4. What are the common methods to handle percentages in algebraic equations?
5. How do logarithmic functions assist in solving exponential equations?

Tip

When dealing with exponential equations, always look for ways to express terms with a common base to simplify the solving process.