$$ \left. \begin{array} { | l | l | } \hline 2.4 & { 2 ^ { x } - 5.2 ^ { x + 1 } = - 144 } \ \hline \end{array} ight. $$

Question

$$ \left. \begin{array} { | l | l | } \hline 2.4 & { 2 ^ { x } - 5.2 ^ { x + 1 } = - 144 } \ \hline \end{array} ight. $$

UpStudy AI · Accepted Answer

To solve the equation:

$$
2^{x} - 5 \cdot 2^{x + 1} = -144
$$

Let's follow the steps systematically.

### Step 1: Simplify the Equation

First, recognize that $$2^{x + 1}$$ can be rewritten using the properties of exponents:

$$
2^{x + 1} = 2^{x} \cdot 2 = 2 \cdot 2^{x}
$$

Substitute this back into the original equation:

$$
2^{x} - 5 \cdot (2 \cdot 2^{x}) = -144
$$

$$
2^{x} - 10 \cdot 2^{x} = -144
$$

### Step 2: Factor Out $$2^{x}$$

Factor $$2^{x}$$ from the left side of the equation:

$$
2^{x}(1 - 10) = -144
$$

$$
2^{x}(-9) = -144
$$

### Step 3: Solve for $$2^{x}$$

Divide both sides by $$-9$$ to isolate $$2^{x}$$:

$$
2^{x} = \frac{-144}{-9} = 16
$$

### Step 4: Solve for $$x$$

Recognize that $$16$$ is a power of $$2$$:

$$
16 = 2^4
$$

Therefore:

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

Since the bases are equal, the exponents must be equal:

$$
x = 4
$$

### **Answer**

$$ x = 4 $$
$$ x = 4 $$

\( \left. \begin{array} { | l | l | } \hline 2.4 & { 2 ^ { x } - 5.2 ^ { x + 1 } = - 144 } \\ \hline \end{array} \right. \)