To solve the expression $\left(-\frac{5}{9}\right)^6$, we will follow these steps:

Step 1: Simplify the expression by applying the exponent

The given expression is $\left(-\frac{5}{9}\right)^6$. When raising a fraction to an exponent, both the numerator and the denominator are raised to that exponent:

$\left(-\frac{5}{9}\right)^6 = \frac{(-5)^6}{9^6}$

Step 2: Calculate the powers

1. Calculate $(-5)^6$:

   • Since the exponent is even, $(-5)^6$ will be positive.
   • $(-5)^6 = (-5) \times (-5) \times (-5) \times (-5) \times (-5) \times (-5) = 15625$.

2. Calculate $9^6$:

   • $9^6 = 9 \times 9 \times 9 \times 9 \times 9 \times 9 = 531441$.

Step 3: Write the result as a fraction

Now, substitute the calculated values into the fraction:

$\left(-\frac{5}{9}\right)^6 = \frac{15625}{531441}$

Final Answer

The simplified value of $\left(-\frac{5}{9}\right)^6$ is $\frac{15625}{531441}$.

So, the final answer is: "The simplified value of $\left(-\frac{5}{9}\right)^6$ is $\frac{15625}{531441}$."

Would you like any further details or have any questions?

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Here are 5 related questions:

1. How would the result change if the exponent were odd instead of even?
2. What is the decimal equivalent of the fraction $\frac{15625}{531441}$?
3. How can you simplify the expression $\left(\frac{5}{9}\right)^6$ without calculating the exact value?
4. What would the answer be if the fraction were $\left(\frac{5}{9}\right)^{-6}$?
5. How does the sign of the base affect the outcome when the exponent is even?

Tip: When raising a negative number to an even power, the result is always positive.