Let's go through the questions step by step and solve them.

1. Evaluate: (-4)^(-3)

Using the formula for negative exponents:
$a^{-n} = \frac{1}{a^n}$
$(-4)^{-3} = \frac{1}{(-4)^3} = \frac{1}{-64} = -\frac{1}{64}$

2. Simplify: $(3^{-7} \cdot 3^{-9}) \cdot 3^{-4}$

Using the property of exponents $a^m \cdot a^n = a^{m+n}$:
$3^{-7} \cdot 3^{-9} \cdot 3^{-4} = 3^{-7-9-4} = 3^{-20}$
The final answer is:
$\boxed{3^{-20}}$

3. Find the value of $(3^7 + 4^{-3} + 5^3)^0$

Anything raised to the power 0 equals 1, as long as the base is not undefined:
$(3^7 + 4^{-3} + 5^3)^0 = 1$

4. Evaluate: $\left[ \left( \frac{1}{2} \right)^{-1} + \left( \frac{1}{3} \right)^{-1} \right]^{-1}$

First, calculate the individual terms:
$\left( \frac{1}{2} \right)^{-1} = 2, \quad \left( \frac{1}{3} \right)^{-1} = 3$
Now add them:
$2 + 3 = 5$
Finally, take the reciprocal:
$5^{-1} = \frac{1}{5}$

The answer is:
$\boxed{\frac{1}{5}}$

5. Express 31860000000 in standard form.

Count the number of digits after the first digit (3):
$31860000000 = 3.186 \times 10^{10}$

6. Find $x$ such that $(-5)^{x+1} \times (-5)^5 = (-5)^7$

Using the property of exponents $a^m \cdot a^n = a^{m+n}$:
$(-5)^{x+1} \cdot (-5)^5 = (-5)^{x+1+5}$
Equating to the right-hand side:
$x+1+5 = 7$
Solve for $x$:
$x = 7 - 6 = 1$

7. Solve $(81)^{-4} \div (729)^{2-x} = 9^{4x}$

Step 1: Express 81 and 729 as powers of 9:
$81 = 9^2, \quad 729 = 9^3$
Substitute these into the equation:
$(9^2)^{-4} \div (9^3)^{2-x} = 9^{4x}$
Simplify exponents:
$9^{-8} \div 9^{3(2-x)} = 9^{4x}$
Use the rule $\frac{a^m}{a^n} = a^{m-n}$:
$9^{-8 - 3(2-x)} = 9^{4x}$
Simplify the exponent:
$-8 - 6 + 3x = 4x$
Solve for $x$:
$-14 + 3x = 4x$
$x = -14$

Final Answers:

1. $-\frac{1}{64}$
2. $3^{-20}$
3. $1$
4. $\frac{1}{5}$
5. $3.186 \times 10^{10}$
6. $x = 1$
7. $x = -14$

Let me know if you'd like a detailed explanation of any step!

5 Related Questions:

1. How do you simplify expressions with negative exponents?
2. What is the significance of standard form in mathematics?
3. How do you solve equations involving fractional exponents?
4. Can you explain when $(a+b)^0$ becomes undefined?
5. How do you convert large numbers into scientific notation?

Tip:

When dealing with exponents, always remember the basic laws of exponents ($a^m \cdot a^n = a^{m+n}$, $(a^m)^n = a^{m \cdot n}$, etc.) to simplify efficiently!