(6) $$ 5^{2+x}-5^{x}=5^{x} \cdot 23+1 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
5^{2 + x} - 5^{x} = 5^{x} \cdot 23 + 1
$$

**Step 1: Simplify the Equation**

First, recognize that $$5^{2 + x}$$ can be rewritten using the properties of exponents:

$$
5^{2 + x} = 5^2 \cdot 5^x = 25 \cdot 5^x
$$

Substitute this back into the original equation:

$$
25 \cdot 5^x - 5^x = 23 \cdot 5^x + 1
$$

**Step 2: Combine Like Terms**

Subtract $$5^x$$ from both sides:

$$
24 \cdot 5^x = 23 \cdot 5^x + 1
$$

Subtract $$23 \cdot 5^x$$ from both sides:

$$
5^x = 1
$$

**Step 3: Solve for $$x$$**

The equation $$5^x = 1$$ holds true when the exponent is zero because any non-zero number raised to the power of zero is 1:

$$
x = 0
$$

**Verification:**

Plug $$x = 0$$ back into the original equation to verify:

$$
5^{2 + 0} - 5^{0} = 25 - 1 = 24
$$

$$
23 \cdot 5^{0} + 1 = 23 \cdot 1 + 1 = 24
$$

Both sides are equal, confirming that $$x = 0$$ is indeed the solution.

**Answer:**  
$$x = 0$$
$$x = 0$$

(6) \( 5^{2+x}-5^{x}=5^{x} \cdot 23+1 \)