Solve the equations:
`2^(x^(2)):2^(x)=8:1`

Answer

To solve the equation \( \frac{2^{x^2}}{2^x} = \frac{8}{1} \), we can follow these steps:

### Step 1: Simplify the Left Side
Using the property of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify the left side:

\[ \frac{2^{x^2}}{2^x} = 2^{x^2 - x} \]

### Step 2: Rewrite the Right Side
Next, we rewrite the right side \( \frac{8}{1} \) as \( 8 \). Since \( 8 = 2^3 \), we can express it as:

\[ \frac{8}{1} = 2^3 \]

### Step 3: Set the Exponents Equal
Now, we have the equation:

\[ 2^{x^2 - x} = 2^3 \]

Since the bases are the same, we can set the exponents equal to each other:

\[ x^2 - x = 3 \]

### Step 4: Rearrange to Form a Quadratic Equation
Rearranging the equation gives us:

\[ x^2 - x - 3 = 0 \]

### Step 5: Apply the Quadratic Formula
To solve the quadratic equation \( x^2 - x - 3 = 0 \), we will use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = -1 \), and \( c = -3 \).

### Step 6: Substitute Values into the Formula
Substituting the values into the formula:

\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \]

This simplifies to:

\[ x = \frac{1 \pm \sqrt{1 + 12}}{2} \]

### Step 7: Simplify Under the Square Root
Calculating the value under the square root gives:

\[ x = \frac{1 \pm \sqrt{13}}{2} \]

### Final Solution
Thus, the solutions for \( x \) are:

\[ x = \frac{1 + \sqrt{13}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{13}}{2} \]