Solve algebraically. a) 4^2x=4^6 b) 5^x=625 c) 3^2x+1=9 d) 10^x+1=10^2x-3 e) 4^3x-2=32^x+1 f) 25^x+1=125^x-2

## Step 1: Solving for x in exponential equations with the same base<br />### When the bases are the same, set the exponents equal to each other and solve for $x$.<br /><br />## Step 2: Solving $4^{2x} = 4^6$<br />### Since the bases are the same, we equate the exponents: $2x = 6$. Dividing both sides by 2 gives $x = 3$.<br /><br />## Step 3: Solving $5^x = 625$<br />### Rewrite 625 as a power of 5: $625 = 5^4$. Thus, $5^x = 5^4$. Equating exponents gives $x = 4$.<br /><br />## Step 4: Solving $3^{2x+1} = 9$<br />### Rewrite 9 as a power of 3: $9 = 3^2$. Thus, $3^{2x+1} = 3^2$. Equating exponents gives $2x + 1 = 2$. Subtracting 1 from both sides gives $2x = 1$. Dividing both sides by 2 gives $x = \frac{1}{2}$.<br /><br />## Step 5: Solving $10^{x+1} = 10^{2x-3}$<br />### Since the bases are the same, we equate the exponents: $x + 1 = 2x - 3$. Subtracting $x$ from both sides gives $1 = x - 3$. Adding 3 to both sides gives $x = 4$.<br /><br />## Step 6: Solving $4^{3x-2} = 32^{x+1}$<br />### Rewrite both sides with a common base of 2. $4 = 2^2$ and $32 = 2^5$. So, $(2^2)^{3x-2} = (2^5)^{x+1}$. This simplifies to $2^{6x-4} = 2^{5x+5}$. Equating exponents gives $6x - 4 = 5x + 5$. Subtracting $5x$ from both sides gives $x - 4 = 5$. Adding 4 to both sides gives $x = 9$.<br /><br />## Step 7: Solving $25^{x+1} = 125^{x-2}$<br />### Rewrite both sides with a common base of 5. $25 = 5^2$ and $125 = 5^3$. So, $(5^2)^{x+1} = (5^3)^{x-2}$. This simplifies to $5^{2x+2} = 5^{3x-6}$. Equating exponents gives $2x + 2 = 3x - 6$. Subtracting $2x$ from both sides gives $2 = x - 6$. Adding 6 to both sides gives $x = 8$.