Which Of The Following Is Equivalent To $36^{-\frac{1}{2}}$?A. $-18$ B. $-6$ C. $\frac{1}{18}$ D. $\frac{1}{6}$

Feb 28, 2025 by ADMIN 119 views

**Simplifying Exponents and Radicals: A Guide to Equivalent Expressions**

Understanding Exponents and Radicals

In mathematics, exponents and radicals are two fundamental concepts that are used to represent complex numbers and expressions. Exponents are used to represent repeated multiplication of a number, while radicals are used to represent the square root of a number. In this article, we will focus on simplifying exponents and radicals, and we will explore the concept of equivalent expressions.

What is an Exponent?

An exponent is a small number that is written above and to the right of a number or expression. It represents the number of times that the base number is multiplied by itself. For example, in the expression $2^3$ , the exponent $3$ represents the number of times that the base number $2$ is multiplied by itself. In this case, $2^3$ is equal to $2 \times 2 \times 2$ , which is equal to $8$ .

What is a Radical?

A radical is a symbol that is used to represent the square root of a number. It is denoted by the symbol $\sqrt{}$ . For example, in the expression $\sqrt{16}$ , the radical symbol represents the square root of $16$ . In this case, $\sqrt{16}$ is equal to $4$ , because $4$ multiplied by $4$ is equal to $16$ .

Simplifying Exponents and Radicals

When simplifying exponents and radicals, we need to follow certain rules and guidelines. One of the most important rules is the rule of exponents, which states that when we multiply two numbers with the same base, we can add their exponents. For example, in the expression $2^3 \times 2^4$ , we can add the exponents to get $2^{3+4}$ , which is equal to $2^7$ .

Another important rule is the rule of radicals, which states that when we multiply two numbers with the same radical, we can multiply their radicands. For example, in the expression $\sqrt{16} \times \sqrt{25}$ , we can multiply the radicands to get $\sqrt{16 \times 25}$ , which is equal to $\sqrt{400}$ .

Simplifying Negative Exponents

Negative exponents are a special type of exponent that can be simplified using certain rules and guidelines. One of the most important rules is the rule of negative exponents, which states that when we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, in the expression $2^{-3}$ , we can rewrite it as $\frac{1}{2^3}$ , which is equal to $\frac{1}{8}$ .

Simplifying Radical Expressions

Radical expressions are a special type of expression that can be simplified using certain rules and guidelines. One of the most important rules is the rule of radical expressions, which states that when we have a radical expression, we can simplify it by finding the square root of the radicand. For example, in the expression $\sqrt{16}$ , we can simplify it by finding the square root of $16$ , which is equal to $4$ .

Which of the Following is Equivalent to $36^{-\frac{1}{2}}$?

Now that we have discussed the concept of exponents and radicals, and we have explored the rules and guidelines for simplifying them, we can now focus on the problem at hand. The problem asks us to find the equivalent expression for $36^{-\frac{1}{2}}$. To solve this problem, we need to follow the rules and guidelines that we have discussed earlier.

First, we need to rewrite the negative exponent as a positive exponent by taking the reciprocal of the base. In this case, we can rewrite $36^{-\frac{1}{2}}$ as $\frac{1}{36^{\frac{1}{2}}}$.

Next, we need to simplify the radical expression by finding the square root of the radicand. In this case, we can simplify $36^{\frac{1}{2}}$ by finding the square root of $36$ , which is equal to $6$ .

Therefore, we can rewrite $36^{-\frac{1}{2}}$ as $\frac{1}{6}$.

Conclusion

In conclusion, we have discussed the concept of exponents and radicals, and we have explored the rules and guidelines for simplifying them. We have also focused on the problem at hand, which asks us to find the equivalent expression for $36^{-\frac{1}{2}}$. By following the rules and guidelines that we have discussed earlier, we have been able to simplify the expression and find the equivalent expression, which is $\frac{1}{6}$.

Answer

Q: What is the difference between an exponent and a radical?

A: An exponent is a small number that is written above and to the right of a number or expression, representing the number of times that the base number is multiplied by itself. A radical, on the other hand, is a symbol that is used to represent the square root of a number.

Q: How do I simplify a negative exponent?

A: To simplify a negative exponent, you can rewrite it as a positive exponent by taking the reciprocal of the base. For example, in the expression $2^{-3}$ , you can rewrite it as $\frac{1}{2^3}$ , which is equal to $\frac{1}{8}$ .

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can find the square root of the radicand. For example, in the expression $\sqrt{16}$ , you can simplify it by finding the square root of $16$ , which is equal to $4$ .

Q: What is the rule of exponents?

A: The rule of exponents states that when you multiply two numbers with the same base, you can add their exponents. For example, in the expression $2^3 \times 2^4$ , you can add the exponents to get $2^{3+4}$ , which is equal to $2^7$ .

Q: What is the rule of radicals?

A: The rule of radicals states that when you multiply two numbers with the same radical, you can multiply their radicands. For example, in the expression $\sqrt{16} \times \sqrt{25}$ , you can multiply the radicands to get $\sqrt{16 \times 25}$ , which is equal to $\sqrt{400}$ .

Q: How do I simplify a complex expression with exponents and radicals?

A: To simplify a complex expression with exponents and radicals, you need to follow the order of operations (PEMDAS):

Evaluate any expressions inside parentheses.
Evaluate any exponential expressions.
Evaluate any radical expressions.
Multiply and divide any remaining expressions.

Q: What is the equivalent expression for $36^{-\frac{1}{2}}$?

A: To find the equivalent expression for $36^{-\frac{1}{2}}$, you can rewrite the negative exponent as a positive exponent by taking the reciprocal of the base. Then, you can simplify the radical expression by finding the square root of the radicand. The equivalent expression is $\frac{1}{6}$ .

Q: How do I know which answer is correct?

A: To determine which answer is correct, you need to follow the rules and guidelines for simplifying exponents and radicals. You can also use a calculator to check your answer.

Q: What are some common mistakes to avoid when simplifying exponents and radicals?

A: Some common mistakes to avoid when simplifying exponents and radicals include:

Forgetting to rewrite negative exponents as positive exponents.
Forgetting to simplify radical expressions.
Not following the order of operations (PEMDAS).
Not checking your answer using a calculator.

Conclusion

In conclusion, we have discussed some frequently asked questions about exponents and radicals, and we have provided answers to help you understand the concepts better. We have also covered some common mistakes to avoid when simplifying exponents and radicals. By following the rules and guidelines, you can simplify complex expressions with exponents and radicals and find the equivalent expression.