Question

what is the value of $9 {}^{6} \times {9}^{4} \\ {9}^{12}$

Answer 1

We will investigate how to simplify fractions involving powers of similar bases.

The following fraction is given for simplification:

$\frac{9^6\cdot9^4}{9^{12}}$

When we have fraction with numerator and denominator have similar digits as basis; however, different powers we apply power rules of multiplication and division as follows:

$\begin{gathered} \text{Multiplication: x}^a\cdot x^b=x^{a+b} \\ \\ Division\text{: }\frac{x^a}{x^b}=x^{a-b} \end{gathered}$

Using the above rules we will simplify the given expression. We will keep in mind the order of priority for mathematical opperations i.e ( PEMDAS ).

We will first multiply out the result in the numerator using the multiplication power rule as follows:

$\begin{gathered} \frac{9^{6+4}}{9^{12}} \\ \\ \frac{9^{10}}{9^{12}} \end{gathered}$

Then we will apply the division power rule to the above resulting expression and simplify further as follows:

$\begin{gathered} \frac{9^{10}}{9^{12}\text{ }}=9^{10-12} \\ \\ 9^{-2}\text{ } \end{gathered}$

Then we will apply the negative exponent power rule. Where any negative exponent can be converted to positive exponent by reciprocating the base as follows:

$\text{Negative Exponent Rule: x}^{-a}\text{ = }\frac{1}{x^a}$

Apply the above rule:

$\begin{gathered} 9^{-2}\text{ = }\frac{1}{9^2} \\ \\ \frac{1}{81}\ldots\text{ Answer} \end{gathered}$