The expression you have uploaded is:

$\left( \frac{s^6 t^{-4}}{9 s^{-1} t} \right)^{-2}$

Let's simplify this step by step:

Step 1: Simplify the expression inside the parentheses.

$\frac{s^6 t^{-4}}{9 s^{-1} t}$

Combine the terms with $s$ and $t$:

• For $s$, we have $s^6$ in the numerator and $s^{-1}$ in the denominator. Using the property of exponents $\frac{a^m}{a^n} = a^{m-n}$, we get:

$s^{6 - (-1)} = s^{6+1} = s^7$

• For $t$, we have $t^{-4}$ in the numerator and $t$ (which is $t^1$) in the denominator. Using the same exponent rule, we get:

$t^{-4 - 1} = t^{-5}$

So the expression becomes:

$\frac{s^7 t^{-5}}{9}$

Step 2: Apply the exponent $-2$.

Now, raise the entire expression to the power of $-2$:

$\left( \frac{s^7 t^{-5}}{9} \right)^{-2}$

We will apply the exponent to each part:

• For $s^7$, $(s^7)^{-2} = s^{7 \times (-2)} = s^{-14}$.
• For $t^{-5}$, $(t^{-5})^{-2} = t^{-5 \times (-2)} = t^{10}$.
• For $9$, $9^{-2} = \frac{1}{9^2} = \frac{1}{81}$.

So, the expression becomes:

$\frac{s^{-14} t^{10}}{1/81}$

Step 3: Simplify the fraction.

Since dividing by $1/81$ is equivalent to multiplying by $81$, we get:

$81 s^{-14} t^{10}$

Finally, to eliminate the negative exponent on $s$, we move it to the denominator:

$\frac{81 t^{10}}{s^{14}}$

Final Answer:

$\frac{81 t^{10}}{s^{14}}$

Would you like further details or have any questions about this simplification?

Here are 5 related questions to explore:

1. How do you simplify more complex fractions involving exponents?
2. What are the general rules for handling negative exponents?
3. How do you distribute exponents over products and quotients?
4. How do you simplify expressions when both numerator and denominator have negative exponents?
5. Can you raise a fraction to a fractional exponent, and how does it affect both the numerator and denominator?

Tip: Always remember that a negative exponent indicates the reciprocal of the base raised to the positive exponent.