Simplify the expression.
[tex]\begin{array}{l}
(x^3)^{-3} \\
\frac{1}{x^{[?]}}
\end{array}[/tex]
Answer (1)
Apply the power of a power rule: ( x a ) b = x a × b .
Calculate the exponent: ( x 3 ) − 3 = x 3 × ( − 3 ) = x − 9 .
Rewrite the expression with a positive exponent: x − 9 = x 9 1 .
Determine the value of [?]: [ ?] = 9 , so the simplified expression is 9 .
Explanation
Understanding the Problem We are asked to simplify the expression ( x 3 ) − 3 and express it in the form x [ ?] 1 where we need to find the value of [?].
Applying the Power of a Power Rule To simplify the expression, we will use the power of a power rule, which states that ( x a ) b = x a × b . In our case, a = 3 and b = − 3 .
Calculating the Exponent Applying the power of a power rule, we have ( x 3 ) − 3 = x 3 × ( − 3 ) = x − 9 .
Rewriting with a Positive Exponent Now, we need to express x − 9 in the form x [ ?] 1 . Recall that x − n = x n 1 . Therefore, x − 9 = x 9 1 .
Finding the Value of [?] Comparing this with x [ ?] 1 we see that [ ?] = 9 .
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with memory sizes (e.g., kilobytes, megabytes, gigabytes, etc.) or in physics when calculating quantities that scale exponentially, like radioactive decay or compound interest. For example, if a computer's memory doubles every year, the memory size can be modeled using exponential growth. Similarly, understanding negative exponents helps in expressing very small quantities, such as the size of atoms or the concentration of pollutants in environmental science. These concepts are also vital in financial mathematics, particularly in calculating compound interest and depreciation.
