Rewrite the following without an exponent: $(-9)^{-2}
Answer (2)
Apply the negative exponent rule: ( − 9 ) − 2 = ( − 9 ) 2 1 .
Evaluate the square: ( − 9 ) 2 = 81 .
Rewrite the expression: 81 1 .
The final answer is 81 1 .
Explanation
Understanding the Problem We are asked to rewrite the expression ( − 9 ) − 2 without using an exponent. This involves understanding how negative exponents work.
Applying the Negative Exponent Rule A negative exponent means we take the reciprocal of the base raised to the positive of that exponent. In other words, a − n = a n 1 . Applying this to our expression, we get: ( − 9 ) − 2 = ( − 9 ) 2 1
Evaluating the Square Now we need to evaluate ( − 9 ) 2 . This means ( − 9 ) × ( − 9 ) . Since a negative times a negative is a positive, we have: ( − 9 ) 2 = ( − 9 ) × ( − 9 ) = 81
Final Result Substituting this back into our expression, we get: ( − 9 ) 2 1 = 81 1
Examples
Understanding negative exponents is crucial in many scientific and engineering applications. For instance, in physics, the force between two charged particles is inversely proportional to the square of the distance between them, described by Coulomb's law. This relationship involves negative exponents, as the force is proportional to r − 2 , where r is the distance. Similarly, in finance, understanding exponential decay, which also uses negative exponents, is essential for calculating the present value of future cash flows or the depreciation of assets. These concepts are not just abstract mathematical ideas but are fundamental tools for modeling and understanding the world around us.
To rewrite ( − 9 ) − 2 without an exponent, use the negative exponent rule to find ( − 9 ) − 2 = ( − 9 ) 2 1 , which simplifies to 81 1 . Therefore, the final answer is 81 1 .
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