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Let's simplify each of the given expressions step by step:

### a. \( x^0 \cdot x^{-5} \cdot x^4 \)
Using the rule for exponents: \( a^m \cdot a^n = a^{m+n} \):
\[
x^0 \cdot x^{-5} \cdot x^4 = x^{0 + (-5) + 4} = x^{-1} = \frac{1}{x}
\]

### b. \( 5^0 \cdot 5^{-2} \cdot 5^4 \)
Similarly, using the rule for exponents:
\[
5^0 \cdot 5^{-2} \cdot 5^4 = 5^{0 + (-2) + 4} = 5^2 = 25
\]

### c. \( (3^{-2})^2 \)
Using the power of a power rule: \( (a^m)^n = a^{m \cdot n} \):
\[
(3^{-2})^2 = 3^{-2 \cdot 2} = 3^{-4} = \frac{1}{3^4} = \frac{1}{81}
\]

### d. \( 4^{-3} \cdot 4^5 \)
Using the same exponent rule as above:
\[
4^{-3} \cdot 4^5 = 4^{-3+5} = 4^2 = 16
\]

### e. \( (-x)^0 \)
Any number raised to the power of 0 is 1, including negative values:
\[
(-x)^0 = 1
\]

### f. \( -x^0 \)
Here, \( x^0 = 1 \), so the expression becomes:
\[
-x^0 = -1
\]

### g. \( 5a^{-8}b^4 \)
This expression cannot be simplified further unless more values are given for the variables, so we leave it as:
\[
5a^{-8}b^4 = \frac{5b^4}{a^8}
\]

### h. \( \frac{ab^{-1}c}{c^{-1}} \)
Simplifying each part:
- \( b^{-1} = \frac{1}{b} \)
- \( c^{-1} = \frac{1}{c} \)
Thus, the expression becomes:
\[
\frac{ab^{-1}c}{c^{-1}} = \frac{ac}{b} \cdot c = \frac{ac^2}{b}
\]

### i. \( \frac{x^{-9}x^4}{x^5} \)
First, simplify the numerator using the exponent rule:
\[
x^{-9} \cdot x^4 = x^{-5}
\]
Now simplify the entire expression:
\[
\frac{x^{-5}}{x^5} = x^{-5-5} = x^{-10} = \frac{1}{x^{10}}
\]

### j. \( \left( \frac{3x^{-1}}{9y^2} \right)^{-2} \)
First, simplify the inside of the parentheses:
\[
\frac{3x^{-1}}{9y^2} = \frac{x^{-1}}{3y^2}
\]
Now apply the exponent \( -2 \) to each term:
\[
\left( \frac{x^{-1}}{3y^2} \right)^{-2} = \frac{(3y^2)^2}{(x^{-1})^2} = \frac{9y^4}{x^2}
\]

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If you have any questions or would like a further breakdown of any steps, feel free to ask!

### Additional Questions:
1. How would you simplify an expression like \( x^a \cdot x^b \cdot x^c \)?
2. What happens when you have negative exponents in both the numerator and denominator?
3. Can you explain the reasoning behind why \( (-x)^0 = 1 \)?
4. How does multiplying powers of the same base with different exponents work in general?
5. How can you simplify expressions with variables raised to a power in both the numerator and denominator?

#### Tip:
Always remember that \( a^0 = 1 \) for any non-zero \( a \), regardless of whether the base is positive or negative.

Simplify the following expressions:

a. x^0 · x^−5 · x^4
b. 5^0 · 5^−2 · 5^4
c. (3^−2)^2
d. 4^−3 · 4^5
e. (−x)^0
f. −x^0
g. 5a^−8b^4
h. (ab^−1c)/(c^−1)
i. (x^−9 · x^4)/x^5
j. (3x^−1 / 9y^2)^−2

Learn how to simplify algebraic expressions using the laws of exponents. This problem set covers key concepts such as zero exponents, negative exponents, and simplification techniques, making it ideal for students in Grades 8-10.

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