If $2^{2x-3} = \frac{1}{8^{x-4}}$ ,then the value of x is:

Q: If $2^{2x-3} = \frac{1}{8^{x-4}}$ ,then the value of x is:

To solve the equation $2^{2x-3} = \frac{1}{8^{x-4}}$, we need to manipulate the equation to find the value of $x$. First, let's rewrite the equation in terms of powers of 2. We know that $8$ can be expressed as $2^3$, so: \[ 8^{x-4} = (2^3)^{x-4} = 2^{3(x-4)} \] Thus, the equation becomes: \[ 2^{2x-3} = \frac{1}{2^{3(x-4)}} \] The right side of the equation can be rewritten using the property of exponents $\frac{1}{a^b} = a^{-b}$: \[ 2^{2x-3} = 2^{-3(x-4)} \] Since the bases are the same, we can equate the exponents: \[ 2x - 3 = -3(x - 4) \] Expanding the right side: \[ 2x - 3 = -3x + 12 \] Now, solve for $x$ by first adding $3x$ to both sides: \[ 2x + 3x - 3 = 12 \] \[ 5x - 3 = 12 \] Add 3 to both sides: \[ 5x = 15 \] Divide both sides by 5: \[ x = 3 \] Therefore, the value of $x$ is 3, which corresponds to Option 4. **Explanation for Incorrect Options:** 1. **Option 1 (-1):** If $x = -1$, substituting into the equation $2^{2(-1)-3} = \frac{1}{8^{-1-4}}$ results in $2^{-5} = 2^{15}$, which is incorrect. 2. **Option 2 (1):** If $x = 1$, substituting into the equation $2^{2(1)-3} = \frac{1}{8^{1-4}}$ results in $2^{-1} = 2^{9}$, which is incorrect. 3. **Option 3 (0):** If $x = 0$, substituting into the equation $2^{2(0)-3} = \frac{1}{8^{0-4}}$ results in $2^{-3} = 2^{12}$, which is incorrect. Thus, the only correct solution is $x = 3$, which is Option 4.

Question

Accepted Answer

To solve the equation $2^{2x-3} = \frac{1}{8^{x-4}}$, we need to manipulate the equation to find the value of $x$.

First, let's rewrite the equation in terms of powers of 2. We know that $8$ can be expressed as $2^3$, so:

$$
8^{x-4} = (2^3)^{x-4} = 2^{3(x-4)}
$$

Thus, the equation becomes:

$$
2^{2x-3} = \frac{1}{2^{3(x-4)}}
$$

The right side of the equation can be rewritten using the property of exponents $\frac{1}{a^b} = a^{-b}$:

$$
2^{2x-3} = 2^{-3(x-4)}
$$

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

$$
2x - 3 = -3(x - 4)
$$

Expanding the right side:

$$
2x - 3 = -3x + 12
$$

Now, solve for $x$ by first adding $3x$ to both sides:

$$
2x + 3x - 3 = 12
$$

$$
5x - 3 = 12
$$

Add 3 to both sides:

$$
5x = 15
$$

Divide both sides by 5:

$$
x = 3
$$

Therefore, the value of $x$ is 3, which corresponds to Option 4.

**Explanation for Incorrect Options:**

1. **Option 1 (-1):** If $x = -1$, substituting into the equation $2^{2(-1)-3} = \frac{1}{8^{-1-4}}$ results in $2^{-5} = 2^{15}$, which is incorrect.

2. **Option 2 (1):** If $x = 1$, substituting into the equation $2^{2(1)-3} = \frac{1}{8^{1-4}}$ results in $2^{-1} = 2^{9}$, which is incorrect.

3. **Option 3 (0):** If $x = 0$, substituting into the equation $2^{2(0)-3} = \frac{1}{8^{0-4}}$ results in $2^{-3} = 2^{12}$, which is incorrect.

Thus, the only correct solution is $x = 3$, which is Option 4.