Simplify the expression.
[tex]$\begin{array}{c}
\left(x^{-6}\right)^3 \\
\frac{1}{x^{[?]}}
\end{array}$[/tex]
Answer (1)
Apply the power of a power rule: ( x − 6 ) 3 = x − 6 ⋅ 3 = x − 18 .
Use the negative exponent rule: x − 18 = x 18 1 .
Identify the missing exponent: The expression simplifies to x 18 1 , so the missing exponent is 18.
The simplified expression is 18 .
Explanation
Understanding the Problem We are given the expression ( x − 6 ) 3 and asked to simplify it to the form \frac{1}{x^{[?]}}} . Our goal is to find the missing exponent.
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 have: ( x − 6 ) 3 = x − 6 ⋅ 3 = x − 18 .
Using the Negative Exponent Rule Now, we need to rewrite x − 18 in the form \frac{1}{x^{[?]}}} . Recall the negative exponent rule, which states that a − n = a n 1 . Applying this rule, we get: x − 18 = x 18 1 .
Finding the Missing Exponent Comparing x 18 1 with \frac{1}{x^{[?]}}} , we can see that the missing exponent is 18.
Final Answer Therefore, the simplified expression is x 18 1 .
Examples
Understanding exponent rules is crucial in many scientific fields. For instance, in physics, when dealing with very large or very small numbers, such as the size of atoms or the distance to stars, we use scientific notation which relies heavily on exponent rules. Simplifying expressions with exponents allows scientists to perform calculations more efficiently and understand the relationships between different physical quantities. For example, when calculating the force between two charged particles, the distance between them is squared in the denominator, and understanding how to manipulate exponents is essential for solving such problems.
