\(\begin{array}{l}\left(x^{-6}\right)^3 \\ \frac{1}{x^{[?]}}\end{array}\)
Answer (1)
Apply the power of a power rule: ( x − 6 ) 3 = x − 6 × 3 .
Simplify the exponent: x − 18 .
Use the negative exponent rule: x − 18 = x 18 1 .
The missing exponent is 18 .
Explanation
Understanding the Problem We are given the expression ( x − 6 ) 3 and asked to express it in the form x [ ?] 1
Objective We need to find the exponent of x in the denominator.
Applying the Power of a Power Rule To simplify the expression, we will use the power of a power rule, which states that ( a m ) n = a m × n . Applying this rule to our expression, we get: ( x − 6 ) 3 = x − 6 × 3 = x − 18
Using the Negative Exponent Rule Now, we need to express x − 18 in the form x [ ?] 1 . To do this, we use the negative exponent rule, which states that a − n = a n 1 . Applying this rule, we have: x − 18 = x 18 1
Finding the Missing Exponent Therefore, the missing exponent is 18.
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with memory sizes (e.g., kilobytes, megabytes, gigabytes) or in physics when calculating quantities that scale exponentially, like radioactive decay. For instance, if a file size doubles every year, the file size after n years can be expressed as 2 n times the initial size. Similarly, the intensity of light decreases exponentially as it passes through a medium, which is described using exponents.
