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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 break it down step by step.

  1. Apply the Power of a Power Property: \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]

  2. Simplify the Exponent in the Second Term: \[ 7^{206-2} = 7^{204} \]

  3. Combine the Exponents: When multiplying the same base, we add the exponents: \[ 7^{-21} \cdot 7^{204} = 7^{-21 + 204} = 7^{183} \]

Now we have: \[ 7^{183} \]

  1. Convert to Positive Exponents: Since the expression is already in positive form due to \(7^{183}\), we can express this as: \[ \frac{1}{7^{-183}} = 7^{183} \]

Finally, the answer, written as an improper fraction, is: \[ 7^{183} = \frac{1}{7^{-183}} \]

Thus, the equivalent expression with only positive exponents is: \[ 7^{183} \]

Since the task is to provide the answer as an improper fraction: \[ \frac{1}{7^{-183}} = 7^{183} \]

Thus, the solution to the expression is: \[ \frac{1}{7^{183}} \]