Simplify the expression below.
$\left(x^{-4}\right)^{-6}$
A. $x^{-24}$
B. $x^{-10}$
C. $x^{10}$
D. $x^{24}$
Answer (1)
The problem requires simplifying an expression with exponents. The power of a power rule, ( a m ) n = a m × n , is applied. Multiplying the exponents, − 4 and − 6 , results in 24 . The simplified expression is x 24 . The final answer is x 24 .
Explanation
Understanding the problem We are asked to simplify the expression ( x − 4 ) − 6 . This involves using the power of a power rule in exponents.
Stating the power of a power rule The power of a power rule states that when you raise a power to another power, you multiply the exponents: ( a m ) n = a m × n .
Applying the rule to the expression Applying this rule to our expression, we have ( x − 4 ) − 6 = x ( − 4 ) × ( − 6 ) .
Calculating the exponent Now, we multiply the exponents: − 4 × − 6 = 24 . Therefore, ( x − 4 ) − 6 = x 24 .
Choosing the correct option Comparing our simplified expression x 24 with the given options, we see that it matches option D.
Examples
Understanding exponent rules is crucial in various fields, such as computer science when dealing with data storage sizes (e.g., kilobytes, megabytes, gigabytes) or in physics when calculating quantities that scale exponentially, like radioactive decay or compound interest. For instance, if a data storage device has a capacity that doubles every year, the storage size can be modeled using exponential growth, and simplifying expressions with exponents becomes essential for predicting future storage capabilities. This skill is also useful in financial mathematics when dealing with compound interest calculations, where the amount of money grows exponentially over time, and simplifying expressions with exponents helps in determining the final amount.
