Simplify the expression.
[tex]\begin{array}{c}
\left(x^{-3}\right)^5 \\
\frac{1}{x^{[?]}}
\end{array}[/tex]
Answer (1)
Apply the power of a power rule: ( x a ) b = x a × b .
Multiply the exponents: ( − 3 ) × 5 = − 15 , so the expression becomes x − 15 .
Rewrite the expression using the property x − n = x n 1 , which gives x 15 1 .
The missing exponent is therefore 15 .
Explanation
Understanding the problem We are asked to simplify the expression ( x − 3 ) 5 and express it in the form x [ ?] 1 where we need 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 ( x a ) b = x a ⋅ b . In our case, a = − 3 and b = 5 .
Simplifying the exponent Now, we multiply the exponents: ( − 3 ) ⋅ 5 = − 15 . Therefore, we have ( x − 3 ) 5 = x − 15 .
Rewriting the expression To express x − 15 in the form x [ ?] 1 we use the property x − n = x n 1 . Thus, x − 15 = x 15 1 .
Finding the missing exponent Therefore, the missing exponent is 15.
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 finance when calculating compound interest. For example, if you invest P dollars at an annual interest rate r compounded n times per year, the amount A you'll have after t years is given by A = P ( 1 + n r ) n t . Simplifying and understanding such exponential expressions helps in financial planning and forecasting.
