c. $$ \sqrt{2^{x+1}}=8^{\prime} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\sqrt{2^{x+1}} = 8
$$

Follow these steps:

1. **Rewrite the Square Root as an Exponent:**
   
   $$
   \sqrt{2^{x+1}} = (2^{x+1})^{1/2}
   $$
   
2. **Simplify the Exponent:**
   
   $$
   (2^{x+1})^{1/2} = 2^{\frac{x+1}{2}}
   $$
   
   So the equation becomes:
   
   $$
   2^{\frac{x+1}{2}} = 8
   $$
   
3. **Express 8 as a Power of 2:**
   
   $$
   8 = 2^3
   $$
   
   Now, the equation is:
   
   $$
   2^{\frac{x+1}{2}} = 2^3
   $$
   
4. **Set the Exponents Equal to Each Other:**
   
   Since the bases are the same, the exponents must be equal:
   
   $$
   \frac{x+1}{2} = 3
   $$
   
5. **Solve for $$ x $$:**
   
   Multiply both sides by 2:
   
   $$
   x + 1 = 6
   $$
   
   Subtract 1 from both sides:
   
   $$
   x = 5
   $$

**Answer:**  
$$ x = 5 $$
$$ x = 5 $$

c. \( \sqrt{2^{x+1}}=8^{\prime} \)

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