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]
