The image contains four math problems under the section "Concept Builder 2." Here are the problems and their solutions:

1. Solve the following equations according to the instructions.

1. Is the equation $5^x \cdot 5^y = 5^{12}$ correct?

   • Since the equation involves multiplication of exponents with the same base (5), you can add the exponents: $5^x \cdot 5^y = 5^{x + y}$ For this to equal $5^{12}$, we must have: $x + y = 12$ Therefore, the equation is correct as long as $x + y = 12$.

2. Find the value of $x$ for $2^{x+2} = 2^5 \cdot 2^3$.

   • Again, with the same base, you add the exponents: $2^5 \cdot 2^3 = 2^{5+3} = 2^8$ Thus, the equation becomes: $2^{x+2} = 2^8$ Since the bases are equal, we can equate the exponents: $x + 2 = 8$ Solving for $x$: $x = 6$

3. Find the value of the expression $\frac{729}{27} \div \frac{125}{25}$.

   • Start by simplifying each fraction: $\frac{729}{27} = 27 \quad \text{and} \quad \frac{125}{25} = 5$ So the expression becomes: $27 \div 5 = \frac{27}{5} = 5.4$

4. If $1 + x + y = \frac{a}{b}$ then find the value of $\frac{a}{b}$.

   • The image is partially cut off, so it is unclear whether there are specific values given for $x$ and $y$. But in a generic sense: $1 + x + y = \frac{a}{b}$ without specific values, $\frac{a}{b}$ is simply $1 + x + y$.

Would you like to explore any specific problem in more detail or have any other questions?

Here are five related questions to think about:

1. What happens to the exponents when the bases are different but the operation is the same?
2. How would the equation change if $5^x \cdot 5^y$ was equal to $125^{12}$?
3. Can you simplify $2^{x+3} = 2^7 \cdot 2^{-2}$ and solve for $x$?
4. How do you handle division of fractions like $\frac{729/125}{27/25}$?
5. How would you find $x$ and $y$ if $\frac{a}{b}$ is given as a specific value?

Tip: When dealing with exponents, always try to simplify the expression by combining like terms before solving. This often makes the problem much easier to handle!