Simplify: $5(5 \sqrt{25})^{-2}$A. $\frac{1}{125}$ B. $ 1 25 \frac{1}{25} 25 1 [/tex] C. 625 D. 3,125
Introduction
In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One of the most common types of simplification is dealing with exponents and radicals. In this article, we will focus on simplifying the expression $5(5 \sqrt{25})^{-2}$. We will break down the problem step by step, using the properties of exponents and radicals to arrive at the final answer.
Understanding the Expression
The given expression is $5(5 \sqrt{25})^{-2}$. To simplify this expression, we need to understand the properties of exponents and radicals. The expression inside the parentheses is $5 \sqrt{25}$. We can simplify this expression by first finding the square root of 25, which is 5. Therefore, $5 \sqrt{25} = 5 \times 5 = 25$.
Simplifying the Expression
Now that we have simplified the expression inside the parentheses, we can focus on the exponent. The exponent is -2, which means we need to take the reciprocal of the expression inside the parentheses and raise it to the power of 2. In other words, we need to find $(25)^{-2}$.
Using the Property of Negative Exponents
To simplify $(25)^{-2}$, we can use the property of negative exponents, which states that $a^{-n} = \frac{1}{a^n}$. Therefore, $(25)^{-2} = \frac{1}{(25)^2}$.
Simplifying the Denominator
The denominator of the expression is $(25)^2$. We can simplify this expression by multiplying 25 by itself. Therefore, $(25)^2 = 25 \times 25 = 625$.
Finding the Final Answer
Now that we have simplified the denominator, we can find the final answer. The expression is $5 \times \frac{1}{625}$. To simplify this expression, we can multiply 5 by the reciprocal of 625, which is $\frac{1}{625}$. Therefore, the final answer is $\frac{5}{625}$.
Reducing the Fraction
The fraction $\frac{5}{625}$ can be reduced by dividing both the numerator and the denominator by their greatest common divisor, which is 5. Therefore, $\frac{5}{625} = \frac{1}{125}$.
Conclusion
In this article, we simplified the expression $5(5 \sqrt{25})^{-2}$ using the properties of exponents and radicals. We broke down the problem step by step, starting with simplifying the expression inside the parentheses and then focusing on the exponent. We used the property of negative exponents to simplify the expression and arrived at the final answer, which is $\frac{1}{125}$. This problem demonstrates the importance of understanding the properties of exponents and radicals in simplifying mathematical expressions.
Common Mistakes to Avoid
When simplifying expressions like $5(5 \sqrt{25})^{-2}$, there are several common mistakes to avoid. One of the most common mistakes is not simplifying the expression inside the parentheses before focusing on the exponent. Another mistake is not using the property of negative exponents to simplify the expression. By avoiding these mistakes, we can ensure that we arrive at the correct answer.
Real-World Applications
Simplifying expressions like $5(5 \sqrt{25})^{-2}$ has several real-world applications. In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. In science and engineering, simplifying expressions is often used to model real-world phenomena and make predictions. By understanding the properties of exponents and radicals, we can simplify complex expressions and arrive at the correct answer.
Practice Problems
To practice simplifying expressions like $5(5 \sqrt{25})^{-2}$, try the following problems:
- Simplify the expression $(2 \sqrt{16})^{-3}$.
- Simplify the expression $(3 \sqrt{9})^{-2}$.
- Simplify the expression $(4 \sqrt{4})^{-1}$.
Q&A: Simplifying Expressions with Exponents and Radicals
**Q: What is the first step in simplifying the expression $5(5 \sqrt25})^{-2}$?** A$. We can simplify this expression by first finding the square root of 25, which is 5. Therefore, $5 \sqrt{25} = 5 \times 5 = 25$.
**Q: What is the property of negative exponents that we can use to simplify the expression $(25)^-2}$?** A = \frac{1}{a^n}$. Therefore, $(25)^{-2} = \frac{1}{(25)^2}$.
Q: How can we simplify the denominator of the expression $\frac{1}{(25)^2}$? A: We can simplify the denominator by multiplying 25 by itself. Therefore, $(25)^2 = 25 \times 25 = 625$.
**Q: What is the final answer to the expression $5 \times \frac1}{625}$?** A{625}$. Therefore, the final answer is $\frac{5}{625}$.
**Q: How can we reduce the fraction $\frac5}{625}$?** A{625} = \frac{1}{125}$.
Q: What are some common mistakes to avoid when simplifying expressions like $5(5 \sqrt{25})^{-2}$? A: Some common mistakes to avoid include not simplifying the expression inside the parentheses before focusing on the exponent, and not using the property of negative exponents to simplify the expression.
**Q: What are some real-world applications of simplifying expressions like $5(5 \sqrt25})^{-2}$?** A)^{-2}$ has several real-world applications, including modeling real-world phenomena and making predictions in science and engineering.
**Q: How can I practice simplifying expressions like $5(5 \sqrt25})^{-2}$?** A)^{-2}$ by trying the following problems:
- Simplify the expression $(2 \sqrt{16})^{-3}$.
- Simplify the expression $(3 \sqrt{9})^{-2}$.
- Simplify the expression $(4 \sqrt{4})^{-1}$.
By practicing these problems, you can improve your skills in simplifying expressions and arrive at the correct answer.
Additional Resources
- For more information on simplifying expressions with exponents and radicals, check out the following resources:
- Khan Academy: Simplifying Expressions with Exponents and Radicals
- Mathway: Simplifying Expressions with Exponents and Radicals
- Wolfram Alpha: Simplifying Expressions with Exponents and Radicals
Conclusion
In this article, we simplified the expression $5(5 \sqrt{25})^{-2}$ using the properties of exponents and radicals. We broke down the problem step by step, starting with simplifying the expression inside the parentheses and then focusing on the exponent. We used the property of negative exponents to simplify the expression and arrived at the final answer, which is $\frac{1}{125}$. This problem demonstrates the importance of understanding the properties of exponents and radicals in simplifying mathematical expressions.