The problem requires us to solve five exponential equations for $x$. The equations are: i. $5^{x+2} = 5^{-1}$ ii. $3^{-x} = 9$ iii. $2^2 = 8^x$ iv. $6^{x+4} = 36$ v. $\frac{1}{4} = 16^x$

1. Problem Description

The problem requires us to solve five exponential equations for $x$. The equations are:

i. $5^{x+2} = 5^{-1}$

ii. $3^{-x} = 9$

iii. $2^2 = 8^x$

iv. $6^{x+4} = 36$

v. $\frac{1}{4} = 16^x$

2. Solution Steps

i. $5^{x+2} = 5^{-1}$

Since the bases are equal, we can equate the exponents:

$x+2 = -1$

$x = -1 - 2$

$x = -3$

ii. $3^{-x} = 9$

We can rewrite 9 as $3^2$:

$3^{-x} = 3^2$

Since the bases are equal, we can equate the exponents:

$-x = 2$

$x = -2$

iii. $2^2 = 8^x$

We can rewrite 8 as $2^3$:

$2^2 = (2^3)^x$

$2^2 = 2^{3x}$

Since the bases are equal, we can equate the exponents:

$2 = 3x$

$x = \frac{2}{3}$

iv. $6^{x+4} = 36$

We can rewrite 36 as $6^2$:

$6^{x+4} = 6^2$

Since the bases are equal, we can equate the exponents:

$x+4 = 2$

$x = 2 - 4$

$x = -2$

v. $\frac{1}{4} = 16^x$

We can rewrite $\frac{1}{4}$ as $4^{-1}$ and 16 as $4^2$:

$4^{-1} = (4^2)^x$

$4^{-1} = 4^{2x}$

Since the bases are equal, we can equate the exponents:

$-1 = 2x$

$x = -\frac{1}{2}$

3. Final Answer

i. $x = -3$

ii. $x = -2$

iii. $x = \frac{2}{3}$

iv. $x = -2$

v. $x = -\frac{1}{2}$