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Answer :
- Apply the power of a power rule: $\left(5^{-4}\right)^{\frac{1}{2}} = 5^{-4 \cdot \frac{1}{2}} = 5^{-2}$.
- Use the definition of negative exponents: $5^{-2} = \frac{1}{5^2}$.
- Calculate $5^2 = 25$.
- Therefore, the final answer is $\boxed{\frac{1}{25}}$.
### Explanation
1. Understanding the Problem
We are asked to evaluate the numerical expression $\left(5^{-4}\right)^{\frac{1}{2}}$. This involves using the power of a power rule and understanding negative exponents. Let's break it down step by step.
2. 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: $(a^m)^n = a^{mn}$. Applying this rule to our expression, we get:
$$\left(5^{-4}\right)^{\frac{1}{2}} = 5^{-4 \cdot \frac{1}{2}} = 5^{-2}$$
So, we have simplified the expression to $5^{-2}$.
3. Understanding Negative Exponents
Now, we need to evaluate $5^{-2}$. A negative exponent means we take the reciprocal of the base raised to the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$. Therefore:
$$5^{-2} = \frac{1}{5^2}$$
4. Calculating the Square
Next, we calculate $5^2$, which means $5 \times 5 = 25$. So, we have:
$$5^{-2} = \frac{1}{25}$$
5. Final Answer
Therefore, the value of the expression $\left(5^{-4}\right)^{\frac{1}{2}}$ is $\frac{1}{25}$.
### Examples
Understanding exponents is crucial in many fields, such as calculating compound interest, where the exponent represents the number of compounding periods. For instance, if you invest money in a bank account with compound interest, the formula to calculate the future value involves raising the interest rate plus one to the power of the number of years. This concept also appears in scientific notation, used to represent very large or very small numbers, like the distance to stars or the size of atoms.
- Use the definition of negative exponents: $5^{-2} = \frac{1}{5^2}$.
- Calculate $5^2 = 25$.
- Therefore, the final answer is $\boxed{\frac{1}{25}}$.
### Explanation
1. Understanding the Problem
We are asked to evaluate the numerical expression $\left(5^{-4}\right)^{\frac{1}{2}}$. This involves using the power of a power rule and understanding negative exponents. Let's break it down step by step.
2. 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: $(a^m)^n = a^{mn}$. Applying this rule to our expression, we get:
$$\left(5^{-4}\right)^{\frac{1}{2}} = 5^{-4 \cdot \frac{1}{2}} = 5^{-2}$$
So, we have simplified the expression to $5^{-2}$.
3. Understanding Negative Exponents
Now, we need to evaluate $5^{-2}$. A negative exponent means we take the reciprocal of the base raised to the positive exponent. In other words, $a^{-n} = \frac{1}{a^n}$. Therefore:
$$5^{-2} = \frac{1}{5^2}$$
4. Calculating the Square
Next, we calculate $5^2$, which means $5 \times 5 = 25$. So, we have:
$$5^{-2} = \frac{1}{25}$$
5. Final Answer
Therefore, the value of the expression $\left(5^{-4}\right)^{\frac{1}{2}}$ is $\frac{1}{25}$.
### Examples
Understanding exponents is crucial in many fields, such as calculating compound interest, where the exponent represents the number of compounding periods. For instance, if you invest money in a bank account with compound interest, the formula to calculate the future value involves raising the interest rate plus one to the power of the number of years. This concept also appears in scientific notation, used to represent very large or very small numbers, like the distance to stars or the size of atoms.
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