Apply the Properties of Integer Exponents to generate equivalent expressions to (7−3)7⋅7206−2 with only positive exponent, then solve the expression. Provide the answer as an improper fraction with no exponents or decimals.

1 answer

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

  1. Simplify \((7^{-3})^7\): Using the property \((a^m)^n = a^{m \cdot n}\), we can rewrite this section:

    \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

  2. Simplify \(7^{206 - 2}\): First, calculate \(206 - 2\):

    \[ 206 - 2 = 204 \]

    Thus,

    \[ 7^{206 - 2} = 7^{204} \]

  3. Combine the two parts: Now we combine \(7^{-21}\) and \(7^{204}\):

    \[ 7^{-21} \cdot 7^{204} \]

    Using the property \(a^m \cdot a^n = a^{m+n}\), we have:

    \[ 7^{-21 + 204} = 7^{183} \]

  4. Write in terms of positive exponent: To have a positive exponent, we express \(7^{183}\) as:

    \[ 7^{183} = \frac{1}{7^{-183}} = \frac{1}{7^{-183}} = \frac{1}{7^{183}} \]

Now we have the simplified expression as \(7^{183}\).

  1. Final result: The expression \(7^{183}\) is already in a clear form for the answer, but to express it as a fraction:

    \[ \text{Final Answer:} \quad \frac{1}{7^{183}} \]

This result is in improper fraction form with no exponents or decimals, and it meets the requirement as specified in the problem statement.