Apply the Properties of Integer Exponents to generate equivalent expressions to (7−3)7 ⋅ 7206−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction.(1 point) (7−3)7 ⋅ 7206−2=

1 answer

To simplify the expression \( (7^{-3})^7 \cdot 7^{206 - 2} \) using the properties of integer exponents, we will apply the laws of exponents step by step.

  1. Applying the power of a power property: \[ (a^m)^n = a^{m \cdot n} \] Thus, we have: \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

  2. Simplifying the second part: \[ 7^{206 - 2} = 7^{204} \]

  3. Combining the expressions: Now we can combine the two parts: \[ 7^{-21} \cdot 7^{204} \]

  4. Applying the product of powers property: \[ a^m \cdot a^n = a^{m+n} \] Therefore: \[ 7^{-21 + 204} = 7^{183} \]

Now we have simplified our expression to \( 7^{183} \).

To express this in terms of an improper fraction: \[ 7^{183} = \frac{7^{183}}{1} \]

So the final answer, expressed as an improper fraction, is: \[ \boxed{\frac{7^{183}}{1}} \]