- The expression \(9^5\) is defined by the base 9 being multiplied by itself 5 times.
- Calculating this expansion, we get \(9 \times 9 \times 9 \times 9 \times 9\).
- Reviewing the options, option c matches this expansion.
- The equivalent expression is \(\boxed{c}\).
Explanation
- Understanding Exponentiation
The objective is to identify which of the provided expressions is equivalent to the power \(9^5\). In mathematics, an exponent \(n^k\) represents the base \(n\) being multiplied by itself \(k\) times. Specifically, \(9^5\) means multiplying the base, 9, by itself 5 times: \(9 \times 9 \times 9 \times 9 \times 9\).
- Evaluating the Given Options
Now, let's analyze each option provided:
- Option a: \(9 \times 9 \times 9 \times 9 \times 9 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5\). This is \(9^5 \times 5^9\), which is clearly not equal to \(9^5\).
- Option b: \(5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5\). This is \(5^9\), which is also not equal to \(9^5\).
- Option c: \(9 \times 9 \times 9 \times 9 \times 9\). This is exactly the definition of \(9^5\), as it shows the base 9 multiplied by itself 5 times.
- Conclusion
Comparing our calculation from the definition to the options, we see that option c matches perfectly. Therefore, the expression equivalent to \(9^5\) is \(9 \times 9 \times 9 \times 9 \times 9\), which corresponds to option c. The final answer is \(\boxed{c}\).
Examples
Understanding exponents is essential in many fields, such as calculating compound interest in finance or determining population growth in biology. For example, if a small culture of bacteria doubles every hour, after 5 hours, the amount of bacteria might be represented by a power of 2, such as \(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\). Mastery of this notation allows us to express very large or small numbers in a compact and readable way.