To simplify the expression \((7^{-3})^7 \cdot 7^{206 - 2}\) using the properties of integer exponents, we can start by applying the rules of exponents:
-
Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)
-
Adding Exponents: \(a^m \cdot a^n = a^{m + n}\)
Let's first simplify each part:
-
Simplifying \((7^{-3})^7\): \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]
-
Simplifying \(7^{206 - 2}\): \[ 7^{206 - 2} = 7^{204} \]
Now, combining these expressions: \[ 7^{-21} \cdot 7^{204} = 7^{-21 + 204} = 7^{183} \]
Now, we will express \(7^{183}\) with only positive exponents. Since it already has a positive exponent, we can keep it as is.
Thus: \[ 7^{183} \]
To express this as an improper fraction, we consider that \(7^{183}\) can also be written in the form: \[ \frac{7^{183}}{1} \]
So the final answer as an improper fraction is: \[ \frac{7^{183}}{1} \]
Thus, the simplified form of the original expression is: \[ \frac{7^{183}}{1} \]
This is the answer in improper fraction format: \[ \frac{7^{183}}{1} \]
1 answer