2. Indien $$ 2^{L x}=9 t $$ gee $$ \frac{3^{x-1}+3^{x+1}}{10.6^{x}} $$ in terme van $$ t $$

Question

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Om die uitdrukking $$\frac{3^{x-1} + 3^{x+1}}{10 \cdot 6^{x}}$$ in terme van $$t$$ uit te druk, volg ons hierdie stappe:

### Gegewe:
$$ 2^{Lx} = 9t $$

Ons wil $$\frac{3^{x-1} + 3^{x+1}}{10 \cdot 6^{x}}$$ omskryf in terme van $$t$$.

### Stap 1: Vereenvoudig die Uitdrukking

Eers vereenvoudig ons die breuk:

$$
\frac{3^{x-1} + 3^{x+1}}{10 \cdot 6^{x}} = \frac{\frac{3^x}{3} + 3^{x} \cdot 3}{10 \cdot (2 \cdot 3)^x}
$$

$$
= \frac{\left(\frac{1}{3} + 3\right) 3^x}{10 \cdot 2^x 3^x}
$$

$$
= \frac{\frac{10}{3} \cdot 3^x}{10 \cdot 2^x \cdot 3^x} = \frac{\frac{10}{3}}{10 \cdot 2^x} = \frac{1}{3 \cdot 2^x}
$$

Dus:

$$
\frac{3^{x-1} + 3^{x+1}}{10 \cdot 6^{x}} = \frac{1}{3 \cdot 2^x}
$$

### Stap 2: Los vir $$2^x$$ in Terme van $$t$$

Gegee dat:

$$
2^{Lx} = 9t
$$

Neem die $$L$$-ste wortel van beide kante:

$$
2^x = (9t)^{\frac{1}{L}}
$$

### Stap 3: Vervang $$2^x$$ in die Vereenvoudigde Uitdrukking

Vervang $$2^x$$ in $$\frac{1}{3 \cdot 2^x}$$:

$$
\frac{1}{3 \cdot 2^x} = \frac{1}{3 \cdot (9t)^{\frac{1}{L}}}
$$

### Finale Uitdrukking in Termen van $$t$$:

$$
\frac{3^{x-1} + 3^{x+1}}{10 \cdot 6^{x}} = \frac{1}{3 \cdot (9t)^{\frac{1}{L}}}
$$

**Dus, die uitdrukking in terme van $$t$$ is:**

$$
\boxed{\,\dfrac{1}{\,3\,\cdot\,\left(9\,t\,\right)^{\,1/L\,}}\,}
$$
$$ \frac{1}{3 \cdot (9t)^{1/L}} $$

2. Indien \( 2^{L x}=9 t \) gee \( \frac{3^{x-1}+3^{x+1}}{10.6^{x}} \) in terme van \( t \)