$$ \left. \begin{array} { l } { \frac { ( p ^ { 3 } ) ^ { x - 1 } \cdot p ^ { - 5 } } { \sqrt[ x ] { p ^ { 3 } } } = 1 } \ { ( x \in N ) } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { \frac { ( p ^ { 3 } ) ^ { x - 1 } \cdot p ^ { - 5 } } { \sqrt[ x ] { p ^ { 3 } } } = 1 } \ { ( x \in N ) } \end{array} ight. $$

UpStudy AI · Accepted Answer

To solve the equation:

$$
\frac{(p^3)^{x - 1} \cdot p^{-5}}{\sqrt[x]{p^3}} = 1 \quad \text{where} \quad x \in \mathbb{N}
$$

**Step 1: Simplify the Expression**

First, simplify each part of the equation using exponent rules.

1. **Numerator:**
   $$
   (p^3)^{x - 1} \cdot p^{-5} = p^{3(x - 1)} \cdot p^{-5} = p^{3x - 3 - 5} = p^{3x - 8}
   $$

2. **Denominator:**
   $$
   \sqrt[x]{p^3} = (p^3)^{1/x} = p^{3/x}
   $$

3. **Combine Numerator and Denominator:**
   $$
   \frac{p^{3x - 8}}{p^{3/x}} = p^{3x - 8 - \frac{3}{x}}
   $$

**Step 2: Set the Exponent to Zero**

For the equation to hold true ($$ p^{\text{exponent}} = 1 $$), the exponent must be zero (assuming $$ p \neq 1 $$).

$$
3x - 8 - \frac{3}{x} = 0
$$

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

Multiply both sides by $$ x $$ to eliminate the fraction:

$$
3x^2 - 8x - 3 = 0
$$

This is a quadratic equation in standard form:

$$
3x^2 - 8x - 3 = 0
$$

**Step 4: Use the Quadratic Formula**

The quadratic formula is $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$, where $$ a = 3 $$, $$ b = -8 $$, and $$ c = -3 $$.

$$
x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 3 \cdot (-3)}}{2 \cdot 3} = \frac{8 \pm \sqrt{64 + 36}}{6} = \frac{8 \pm \sqrt{100}}{6} = \frac{8 \pm 10}{6}
$$

This gives two potential solutions:

1. $$ x = \frac{18}{6} = 3 $$
2. $$ x = \frac{-2}{6} = -\frac{1}{3} $$

**Step 5: Consider the Domain**

Since $$ x \in \mathbb{N} $$ (natural numbers), the only valid solution is:

$$
x = 3
$$

**Conclusion**

The natural number solution to the equation is:

**Answer:** $$x = 3$$
$$ x = 3 $$

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