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

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 how to find equivalent expressions.

What is an Exponent?

An exponent is a small number that is written above and to the right of a number, indicating how many times the number should be multiplied by itself. For example, in the expression $2^3$, the exponent $3$ indicates that the number $2$ should be multiplied by itself three times: $2^3 = 2 \times 2 \times 2 = 8$.

What is a Radical?

A radical is a symbol that is used to represent the square root of a number. The radical symbol is denoted by $\sqrt{}$, and it is used to indicate that the number inside the symbol should be divided by itself. For example, in the expression $\sqrt{16}$, the radical symbol indicates that the number $16$ should be divided by itself: $\sqrt{16} = \frac{16}{16} = 1$.

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 negative exponents. According to this rule, a negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, in the expression $2^{-3}$, the negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Simplifying Negative Exponents

To simplify a negative exponent, we need to take the reciprocal of the base and change the sign of the exponent. For example, in the expression $36^{-\frac{1}{2}}$, we can simplify the negative exponent by taking the reciprocal of the base and changing the sign of the exponent: $36^{-\frac{1}{2}} = \frac{1}{36^{\frac{1}{2}}} = \frac{1}{\sqrt{36}} = \frac{1}{6}$.

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

Now that we have simplified the negative exponent, we can compare the simplified expression to the answer choices. The simplified expression is $\frac{1}{6}$, which is equivalent to answer choice D.

Conclusion

In conclusion, simplifying exponents and radicals is an important concept in mathematics that requires a deep understanding of the rules and guidelines. By following the rules of negative exponents and simplifying negative exponents, we can rewrite complex expressions in a simpler form. In this article, we have explored how to simplify the expression $36^{-\frac{1}{2}}$ and have found that it is equivalent to answer choice D, $\frac{1}{6}$.

Answer

Q: What is the rule of negative exponents?

A: The rule of negative exponents states that a negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, in the expression $2^{-3}$, the negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: How do I simplify a negative exponent?

A: To simplify a negative exponent, you need to take the reciprocal of the base and change the sign of the exponent. For example, in the expression $36^{-\frac{1}{2}}$, we can simplify the negative exponent by taking the reciprocal of the base and changing the sign of the exponent: $36^{-\frac{1}{2}} = \frac{1}{36^{\frac{1}{2}}} = \frac{1}{\sqrt{36}} = \frac{1}{6}$.

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, indicating how many times the number should be multiplied by itself. A radical, on the other hand, is a symbol that is used to represent the square root of a number. The radical symbol is denoted by $\sqrt{}$, and it is used to indicate that the number inside the symbol should be divided by itself.

Q: How do I simplify a radical?

A: To simplify a radical, you need to find the largest perfect square that divides the number inside the radical symbol. For example, in the expression $\sqrt{16}$, the largest perfect square that divides $16$ is $4$, so we can simplify the radical as follows: $\sqrt{16} = \sqrt{4 \times 4} = \sqrt{4} \times \sqrt{4} = 2 \times 2 = 4$.

Q: What is the order of operations for simplifying exponents and radicals?

A: The order of operations for simplifying exponents and radicals is as follows:

1. Simplify any negative exponents by taking the reciprocal of the base and changing the sign of the exponent.
2. Simplify any radicals by finding the largest perfect square that divides the number inside the radical symbol.
3. Evaluate any exponential expressions by multiplying the base by itself as many times as indicated by the exponent.
4. Simplify any resulting expressions by combining like terms.

Q: Can you provide an example of how to simplify an expression using the order of operations?

A: Let's consider the expression $2^{-3} \times \sqrt{16}$. To simplify this expression, we need to follow the order of operations:

1. Simplify the negative exponent: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.
2. Simplify the radical: $\sqrt{16} = \sqrt{4 \times 4} = \sqrt{4} \times \sqrt{4} = 2 \times 2 = 4$.
3. Evaluate the exponential expression: $2^{-3} = \frac{1}{8}$.
4. Simplify the resulting expression: $\frac{1}{8} \times 4 = \frac{1}{2}$.

Therefore, the simplified expression is $\frac{1}{2}$.

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:

• Failing to simplify negative exponents by taking the reciprocal of the base and changing the sign of the exponent.
• Failing to simplify radicals by finding the largest perfect square that divides the number inside the radical symbol.
• Failing to evaluate exponential expressions by multiplying the base by itself as many times as indicated by the exponent.
• Failing to simplify resulting expressions by combining like terms.

By avoiding these common mistakes, you can ensure that you are simplifying exponents and radicals correctly and accurately.