Solved on Dec 04, 2023

Solve the equation $\frac{1}{16^{2x+1}} = 16$ for $x$ .

STEP 1

Assumptions
1. The equation is $\frac{1}{16^{2 x+1}}=16$
2. We are solving for $x$

STEP 2

First, we can rewrite the equation to make it easier to solve. We can write $16$ as $16^{1}$ and $16^{2x+1}$ as $16^{-(2x+1)}$ .
$16^{-(2x+1)} = 16^{1}$

STEP 3

Since the bases are the same, we can set the exponents equal to each other.
$-(2x+1) = 1$

STEP 4

Solve the equation for $x$ by first adding $(2x+1)$ to both sides to get rid of the negative sign.
$2x+1 = -1$

STEP 5

Subtract $1$ from both sides to isolate $2x$ on the left side of the equation.
$2x = -1 - 1$

STEP 6

Calculate the right side of the equation.
$2x = -2$

STEP 7

Finally, divide both sides by $2$ to solve for $x$ .
$x = \frac{-2}{2}$

STEP 8

Calculate the value of $x$ .
$x = -1$
So, $x = -1$ is the solution to the equation $\frac{1}{16^{2 x+1}}=16$ .

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Solve the equation $\frac{1}{16^{2x+1}} = 16$ for $x$ .

STEP 1

Assumptions
1. The equation is $\frac{1}{16^{2 x+1}}=16$
2. We are solving for $x$

STEP 2

First, we can rewrite the equation to make it easier to solve. We can write $16$ as $16^{1}$ and $16^{2x+1}$ as $16^{-(2x+1)}$ .
$16^{-(2x+1)} = 16^{1}$

STEP 3

Since the bases are the same, we can set the exponents equal to each other.
$-(2x+1) = 1$

STEP 4

Solve the equation for $x$ by first adding $(2x+1)$ to both sides to get rid of the negative sign.
$2x+1 = -1$

STEP 5

Subtract $1$ from both sides to isolate $2x$ on the left side of the equation.
$2x = -1 - 1$

STEP 6

Calculate the right side of the equation.
$2x = -2$

STEP 7

Finally, divide both sides by $2$ to solve for $x$ .
$x = \frac{-2}{2}$

STEP 8

Calculate the value of $x$ .
$x = -1$
So, $x = -1$ is the solution to the equation $\frac{1}{16^{2 x+1}}=16$ .

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Solve the equation 1162x+1=16\frac{1}{16^{2x+1}} = 16162x+11​=16 for xxx.

STEP 1

Assumptions1. The equation is 1162x+1=16\frac{1}{16^{2 x+1}}=16162x+11​=162. We are solving for xxx

STEP 2

First, we can rewrite the equation to make it easier to solve. We can write 161616 as 16116^{1}161 and 162x+116^{2x+1}162x+1 as 16−(2x+1)16^{-(2x+1)}16−(2x+1).16−(2x+1)=16116^{-(2x+1)} = 16^{1}16−(2x+1)=161

STEP 3

Since the bases are the same, we can set the exponents equal to each other.−(2x+1)=1-(2x+1) = 1−(2x+1)=1

STEP 4

Solve the equation for xxx by first adding (2x+1)(2x+1)(2x+1) to both sides to get rid of the negative sign.2x+1=−12x+1 = -12x+1=−1

STEP 5

Subtract 111 from both sides to isolate 2x2x2x on the left side of the equation.2x=−1−12x = -1 - 12x=−1−1

STEP 6

Calculate the right side of the equation.2x=−22x = -22x=−2

STEP 7

Finally, divide both sides by 222 to solve for xxx.x=−22x = \frac{-2}{2}x=2−2​

STEP 8

Calculate the value of xxx.x=−1x = -1x=−1So, x=−1x = -1x=−1 is the solution to the equation 1162x+1=16\frac{1}{16^{2 x+1}}=16162x+11​=16.

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Solve the equation $\frac{1}{16^{2x+1}} = 16$ for $x$ .

Assumptions
1. The equation is $\frac{1}{16^{2 x+1}}=16$
2. We are solving for $x$

First, we can rewrite the equation to make it easier to solve. We can write $16$ as $16^{1}$ and $16^{2x+1}$ as $16^{-(2x+1)}$ .
$16^{-(2x+1)} = 16^{1}$

Since the bases are the same, we can set the exponents equal to each other.
$-(2x+1) = 1$

Solve the equation for $x$ by first adding $(2x+1)$ to both sides to get rid of the negative sign.
$2x+1 = -1$

Subtract $1$ from both sides to isolate $2x$ on the left side of the equation.
$2x = -1 - 1$

Calculate the right side of the equation.
$2x = -2$

Finally, divide both sides by $2$ to solve for $x$ .
$x = \frac{-2}{2}$

Calculate the value of $x$ .
$x = -1$
So, $x = -1$ is the solution to the equation $\frac{1}{16^{2 x+1}}=16$ .