Okay, I will help you with question 42 from the image. Question 42: For what value of $y$ is $125^y = \frac{1}{25}$? Step 1: Express both sides of the equation with the same base. We can express both $125$ and $25$ as powers of $5$. $$ 125 = 5^3 $$ $$ 25 = 5^2 $$ Substitute these into the equation: $$ (5^3)^y = \frac{1}{5^2} $$ Step 2: Apply exponent rules. Recall the exponent rule $(a^m)^n = a^{mn}$. $$ 5^{3y} = \frac{1}{5^2} $$ Recall the exponent rule $\frac{1}{a^m} = a^{-m}$. $$ 5^{3y} = 5^{-2} $$ Step 3: Equate the exponents. Since the bases are the same, the exponents must be equal. $$ 3y = -2 $$ Step 4: Solve for $y$. Divide both sides by $3$: $$ y = -\frac{2}{3} $$ The value of $y$ is: $$ \boxed{-\frac{2}{3}} $$