Which applies the power of a power rule properly to simplify the expression $\left(7^{10}\right)^5 ?$
A. $\left(7^{10}\right)^5=7^{10+5}=7^2$
B. $\left(7^{10}\right)^5=7^{10-5}=7^5$
C. $\left(7^{10}\right)^5=7^{10+5}=7^{15}$
D. $\left(7^{10}\right)^5=7^{10 \cdot 5}=7^{50}$
Answer (1)
The power of a power rule states that ( a m ) n = a m ⋅ n .
Apply the rule to the expression ( 7 10 ) 5 , where a = 7 , m = 10 , and n = 5 .
Multiply the exponents: 10 ⋅ 5 = 50 .
The simplified expression is 7 50 .
Explanation
Understanding the Power of a Power Rule We are asked to simplify the expression ( 7 10 ) 5 using 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.
Stating the Rule The power of a power rule is given by the formula: ( a m ) n = a m ⋅ n where a is the base and m and n are the exponents.
Applying the Rule to the Expression In our case, we have a = 7 , m = 10 , and n = 5 . Applying the power of a power rule, we get: ( 7 10 ) 5 = 7 10 ⋅ 5 = 7 50
Final Answer Therefore, the correct simplification of the expression ( 7 10 ) 5 is 7 50 .
Examples
The power of a power rule is useful in various fields, such as computer science when dealing with data storage or network bandwidth. For example, if a server has 2 10 kilobytes of memory and you want to determine the total memory in bits, and each kilobyte has 2 3 bits, you would calculate ( 2 10 ) 3 = 2 30 bits. This rule helps simplify calculations involving exponential growth or decay, making it easier to understand and manage large quantities.
