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: $\left( \frac{x^{-1}}{3y^2} \right)^{-2} = \frac{(3y^2)^2}{(x^{-1})^2} = \frac{9y^4}{x^2}$

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.