Simplify the expression. [tex]$\left(\frac{1}{x}\right)^{-3}$[/tex]
Answer (1)
Apply the rule for negative exponents on fractions: ( b a ) − n = ( a b ) n .
Rewrite the expression: ( x 1 ) − 3 = ( 1 x ) 3 .
Simplify the fraction: 1 x = x .
Simplify the expression: ( 1 x ) 3 = x 3 . The final answer is x 3 .
Explanation
Understanding the Problem We are asked to simplify the expression ( x 1 ) − 3 . This involves understanding how negative exponents work, especially when applied to fractions.
Applying the Negative Exponent Rule Recall the property of exponents that states ( b a ) − n = ( a b ) n . Applying this property to our expression, we get: ( x 1 ) − 3 = ( 1 x ) 3
Simplifying the Expression Now, simplify the expression. Since 1 x = x , we have: ( 1 x ) 3 = x 3 Thus, the simplified expression is x 3 .
Final Answer Therefore, the simplified form of the given expression ( x 1 ) − 3 is x 3 .
Examples
Imagine you are designing a system where the size of something is inversely proportional to a certain power of a variable. For example, the intensity of light might be inversely proportional to the cube of the distance from the source. If you initially express this relationship as ( d 1 ) 3 where d is the distance, simplifying it to d − 3 or, equivalently, d 3 1 helps in calculations and understanding how quickly the intensity drops as distance increases. Similarly, ( x 1 ) − 3 can represent a reciprocal relationship raised to a negative power, which simplifies to x 3 , showing a direct cubic relationship.
