The problem asks to simplify the given expressions and write them in positive index form. This means we need to eliminate any negative exponents.

1. Problem Description

The problem asks to simplify the given expressions and write them in positive index form. This means we need to eliminate any negative exponents.

2. Solution Steps

(a) $3^{-2}$

We use the formula $a^{-n} = \frac{1}{a^n}$.

$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

(b) $5^{-1}$

We use the formula $a^{-n} = \frac{1}{a^n}$.

$5^{-1} = \frac{1}{5^1} = \frac{1}{5}$.

(c) $x^{-2} \times x^{-3}$

We use the formula $a^m \times a^n = a^{m+n}$.

$x^{-2} \times x^{-3} = x^{-2 + (-3)} = x^{-5}$.

Now we use the formula $a^{-n} = \frac{1}{a^n}$.

$x^{-5} = \frac{1}{x^5}$.

(d) $a^3 \div a^4$

We use the formula $a^m \div a^n = a^{m-n}$.

$a^3 \div a^4 = a^{3-4} = a^{-1}$.

Now we use the formula $a^{-n} = \frac{1}{a^n}$.

$a^{-1} = \frac{1}{a}$.

(e) $\frac{a^{-1} \times a^{-2}}{a^{-2} \times a^3}$

First, simplify the numerator and the denominator separately using $a^m \times a^n = a^{m+n}$.

Numerator: $a^{-1} \times a^{-2} = a^{-1 + (-2)} = a^{-3}$.

Denominator: $a^{-2} \times a^3 = a^{-2 + 3} = a^1 = a$.

So the expression becomes $\frac{a^{-3}}{a}$.

Now we use the formula $a^m \div a^n = a^{m-n}$.

$\frac{a^{-3}}{a} = \frac{a^{-3}}{a^1} = a^{-3-1} = a^{-4}$.

Now we use the formula $a^{-n} = \frac{1}{a^n}$.

$a^{-4} = \frac{1}{a^4}$.

(f) $a^2 b^{-3}$

$a^2 b^{-3} = a^2 \times \frac{1}{b^3} = \frac{a^2}{b^3}$.

3. Final Answer

(a) $\frac{1}{9}$

(b) $\frac{1}{5}$

(c) $\frac{1}{x^5}$

(d) $\frac{1}{a}$

(e) $\frac{1}{a^4}$

(f) $\frac{a^2}{b^3}$