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Answer :
- The problem involves simplifying an expression with a negative exponent.
- Apply the rule $a^{-n} = \frac{1}{a^n}$ to rewrite the expression as $-\frac{1}{5^2}$.
- Calculate $5^2 = 25$, so the expression becomes $-\frac{1}{25}$.
- The final simplified expression is $\boxed{-\frac{1}{25}}$.
### Explanation
1. Understanding the Problem
We are asked to simplify the expression $-5^{-2}$. This involves understanding negative exponents.
2. Applying the Negative Exponent
Recall that a negative exponent means we take the reciprocal of the base raised to the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$. However, it is important to note the order of operations. The exponent only applies to what is directly to its left. In this case, the exponent -2 applies only to 5, not to -5. So we have $$-5^{-2} = - (5^{-2}) = - \frac{1}{5^2}$$
3. Calculating the Square
Now we need to calculate $5^2$, which means $5 \times 5 = 25$. Therefore, we have $$- \frac{1}{5^2} = - \frac{1}{25}$$
4. Final Answer
So, the simplified expression is $-\frac{1}{25}$. Comparing this to the given options, we see that it matches option C.
### Examples
Understanding negative exponents is crucial in various scientific and engineering fields. For instance, in physics, dealing with very small quantities like the mass of an electron ($9.109 \times 10^{-31}$ kg) requires using negative exponents. Similarly, in computer science, understanding negative exponents helps in representing fractions of memory or processing power.
- Apply the rule $a^{-n} = \frac{1}{a^n}$ to rewrite the expression as $-\frac{1}{5^2}$.
- Calculate $5^2 = 25$, so the expression becomes $-\frac{1}{25}$.
- The final simplified expression is $\boxed{-\frac{1}{25}}$.
### Explanation
1. Understanding the Problem
We are asked to simplify the expression $-5^{-2}$. This involves understanding negative exponents.
2. Applying the Negative Exponent
Recall that a negative exponent means we take the reciprocal of the base raised to the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$. However, it is important to note the order of operations. The exponent only applies to what is directly to its left. In this case, the exponent -2 applies only to 5, not to -5. So we have $$-5^{-2} = - (5^{-2}) = - \frac{1}{5^2}$$
3. Calculating the Square
Now we need to calculate $5^2$, which means $5 \times 5 = 25$. Therefore, we have $$- \frac{1}{5^2} = - \frac{1}{25}$$
4. Final Answer
So, the simplified expression is $-\frac{1}{25}$. Comparing this to the given options, we see that it matches option C.
### Examples
Understanding negative exponents is crucial in various scientific and engineering fields. For instance, in physics, dealing with very small quantities like the mass of an electron ($9.109 \times 10^{-31}$ kg) requires using negative exponents. Similarly, in computer science, understanding negative exponents helps in representing fractions of memory or processing power.
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