What is the value of the expression $(-5)^{-3}$?
Answer (1)
Apply the negative exponent rule: ( − 5 ) − 3 = ( − 5 ) 3 1 .
Calculate ( − 5 ) 3 = ( − 5 ) × ( − 5 ) × ( − 5 ) = − 125 .
Substitute the result: ( − 5 ) − 3 = − 125 1 .
Simplify: x = − 125 1 = − 0.008 .
Explanation
Understanding Negative Exponents We are asked to find the value of the expression ( − 5 ) − 3 . To do this, we need to understand how negative exponents work. 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 the Rule Applying the negative exponent rule, we have ( − 5 ) − 3 = ( − 5 ) 3 1 . Now we need to evaluate ( − 5 ) 3 . This means ( − 5 ) × ( − 5 ) × ( − 5 ) .
Calculating the Cube Let's calculate ( − 5 ) × ( − 5 ) × ( − 5 ) . First, ( − 5 ) × ( − 5 ) = 25 . Then, 25 × ( − 5 ) = − 125 . So, ( − 5 ) 3 = − 125 .
Final Calculation Now we can substitute this back into our expression: ( − 5 ) − 3 = ( − 125 ) 1 . This simplifies to − 125 1 . As a decimal, this is equal to -0.008.
The Answer Therefore, the value of the expression ( − 5 ) − 3 is − 125 1 or -0.008. So, x = − 125 1 = − 0.008 .
Examples
Understanding negative exponents is crucial in many areas, such as calculating very small quantities in science or engineering. For example, when dealing with measurements at the nanoscale, we often encounter numbers like 1 0 − 9 meters (nanometers). Similarly, in computer science, understanding exponents helps in grasping concepts like binary representation and data storage sizes, where values are often expressed as powers of 2 (e.g., 2 − 10 for kilobytes). This knowledge is also useful in finance when calculating depreciation or compound interest rates over time.
