Simplify $\left(7^5\right)^3$
A. $\left(\frac{1}{7}\right)^{15}$
B. $35^3$
C. $7^8$
D. $7^{15}$
Answer (2)
Apply the power of a power rule: ( a m ) n = a m .
Simplify the expression: ( 7 5 ) 3 = 7 53 = 7 15 .
Compare the simplified expression with the given options.
The correct answer is 7 15 .
Explanation
Understanding the Problem We are given the expression ( 7 5 ) 3 . Our goal is to simplify this expression using the power of a power rule.
Applying 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. In other words, ( a m ) n = a m ⋅ n .
Simplifying the Expression Applying this rule to our expression, we get:
( 7 5 ) 3 = 7 5 ⋅ 3 = 7 15
So, the simplified expression is 7 15 .
Choosing the Correct Option Now, we compare our simplified expression 7 15 with the given options:
A. ( 7 1 ) 15 B. 3 5 3 C. 7 8 D. 7 15
We can see that option D matches our simplified expression.
Final Answer Therefore, the correct answer is 7 15 .
Examples
Understanding exponent rules like the power of a power rule is crucial in various fields, such as calculating compound interest or analyzing exponential growth in populations. For instance, if a bacteria population triples every hour, after 5 hours, the population will be 3 5 times the initial population. If we observe this population for 3 days (72 hours), the total growth relative to the initial population is ( 3 5 ) 72 = 3 360 . This illustrates how exponential growth, governed by these rules, can lead to significant changes over time.
The expression ( 7 5 ) 3 simplifies to 7 15 using the power of a power rule. Therefore, the correct answer is option D, 7 15 .
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