Answer :

Sure, let's work through this step-by-step to simplify the given expression:

i) [tex]\(\left[\left(-\frac{2}{9}\right)^3 \times \left(-\frac{2}{9}\right)^7\right] \div \left(-\frac{2}{9}\right)^9\)[/tex]

First, recall the properties of exponents:

1. [tex]\((a^m \times a^n) = a^{m+n}\)[/tex]
2. [tex]\(\left(\frac{a}{b}\right)^m \div \left(\frac{a}{b}\right)^n = \left(\frac{a}{b}\right)^{m-n}\)[/tex]

Let's apply these properties step-by-step.

Step 1: Simplify the multiplication inside the brackets

[tex]\[ \left(-\frac{2}{9}\right)^3 \times \left(-\frac{2}{9}\right)^7 = \left(-\frac{2}{9}\right)^{3+7} = \left(-\frac{2}{9}\right)^{10} \][/tex]

Step 2: Simplify the division

[tex]\[ \left[\left(-\frac{2}{9}\right)^{10}\right] \div \left(-\frac{2}{9}\right)^9 = \left(-\frac{2}{9}\right)^{10-9} = \left(-\frac{2}{9}\right)^1 \][/tex]

So, the expression simplifies to:

[tex]\[ \left(-\frac{2}{9}\right)^1 = -\frac{2}{9} \][/tex]

Therefore, the simplified form of the given expression is:

[tex]\[ -\frac{2}{9} \][/tex]