Which Is Equivalent To $64^{\frac{1}{4}}$?A. $2 \sqrt[4]{4}$ B. 4 C. 16 D. $ 16 4 4 16 \sqrt[4]{4} 16 4 4 [/tex]
Introduction
In mathematics, exponents and roots are fundamental concepts that help us simplify complex expressions and solve equations. When dealing with exponents and roots, it's essential to understand the properties and rules that govern their behavior. In this article, we'll explore the concept of equivalent expressions, focusing on the simplification of exponents and roots. We'll examine the given expression $64^{\frac{1}{4}}$ and determine which of the provided options is equivalent to it.
Understanding Exponents and Roots
Before we dive into the simplification process, let's review the basics of exponents and roots.
- Exponents: An exponent is a small number that is written above and to the right of a base number. It represents the number of times the base number is multiplied by itself. For example, $2^3 = 2 \times 2 \times 2 = 8$.
- Roots: A root is the inverse operation of an exponent. It represents the number that, when raised to a certain power, gives a specific value. For example, $\sqrt[3]{8} = 2$ because $2^3 = 8$.
Simplifying Exponents
Now that we've reviewed the basics, let's focus on simplifying exponents. The given expression $64^{\frac{1}{4}}$ can be simplified using the following steps:
- Break down the exponent: We can break down the exponent $\frac1}{4}$ into a product of two fractions{4} = \frac{1}{2} \times \frac{1}{2}$.
- Apply the exponent rule: Using the exponent rule $a^m \times a^n = a^{m+n}$, we can rewrite the expression as $64^{\frac{1}{2} \times \frac{1}{2}} = (64{\frac{1}{2}}){\frac{1}{2}}$.
- Simplify the inner expression: The inner expression $64^{\frac{1}{2}}$ can be simplified as $\sqrt{64} = 8$.
- Apply the exponent rule again: Using the exponent rule $a^m \times a^n = a^{m+n}$, we can rewrite the expression as $(8)^{\frac{1}{2}} = \sqrt{8}$.
Simplifying Roots
Now that we've simplified the exponent, let's focus on simplifying the root. The expression $\sqrt{8}$ can be simplified using the following steps:
- Break down the root: We can break down the root $\sqrt8}$ into a product of two square roots = \sqrt{4 \times 2}$.
- Apply the root rule: Using the root rule $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can rewrite the expression as $\sqrt{4} \times \sqrt{2}$.
- Simplify the square roots: The square roots $\sqrt{4}$ and $\sqrt{2}$ can be simplified as $2$ and $\sqrt{2}$, respectively.
- Multiply the simplified square roots: The simplified square roots $2$ and $\sqrt{2}$ can be multiplied as $2 \times \sqrt{2} = 2\sqrt{2}$.
Conclusion
In conclusion, the expression $64^\frac{1}{4}}$ can be simplified as $2\sqrt{2}$. This is equivalent to option A$. However, we can further simplify the expression by recognizing that $\sqrt{2}$ is equivalent to $\sqrt[4]{4}$.
Final Answer
Q: What is the difference between an exponent and a root?
A: An exponent is a small number that is written above and to the right of a base number, representing the number of times the base number is multiplied by itself. A root, on the other hand, is the inverse operation of an exponent, representing the number that, when raised to a certain power, gives a specific value.
Q: How do I simplify an exponent?
A: To simplify an exponent, you can break it down into a product of two fractions, apply the exponent rule, and then simplify the inner expression. For example, to simplify $64^{\frac{1}{4}}$, you can break it down into $64^{\frac{1}{2} \times \frac{1}{2}}$, apply the exponent rule, and then simplify the inner expression as $\sqrt{64} = 8$.
Q: How do I simplify a root?
A: To simplify a root, you can break it down into a product of two square roots, apply the root rule, and then simplify the square roots. For example, to simplify $\sqrt{8}$, you can break it down into $\sqrt{4 \times 2}$, apply the root rule, and then simplify the square roots as $2 \times \sqrt{2} = 2\sqrt{2}$.
Q: What is the relationship between exponents and roots?
A: Exponents and roots are inverse operations. An exponent represents the number of times a base number is multiplied by itself, while a root represents the number that, when raised to a certain power, gives a specific value. For example, $2^3 = 8$ and $\sqrt[3]{8} = 2$.
Q: How do I determine which option is equivalent to a given expression?
A: To determine which option is equivalent to a given expression, you can simplify the expression using the rules of exponents and roots, and then compare it to the options. For example, to determine which option is equivalent to $64^{\frac{1}{4}}$, you can simplify it as $2\sqrt{2}$ and then compare it to the options.
Q: What are some common mistakes to avoid when simplifying exponents and roots?
A: Some common mistakes to avoid when simplifying exponents and roots include:
- Not breaking down the exponent or root into smaller parts
- Not applying the exponent or root rule correctly
- Not simplifying the inner expression or square roots
- Not comparing the simplified expression to the options
Q: How can I practice simplifying exponents and roots?
A: You can practice simplifying exponents and roots by working through examples and exercises, such as simplifying $64^{\frac{1}{4}}$ or $\sqrt{8}$. You can also try simplifying more complex expressions and roots, such as $2^{\frac{3}{4}}$ or $\sqrt[3]{27}$.
Conclusion
In conclusion, simplifying exponents and roots is an essential skill in mathematics. By understanding the rules of exponents and roots, you can simplify complex expressions and roots, and determine which option is equivalent to a given expression. Remember to break down the exponent or root into smaller parts, apply the exponent or root rule correctly, and simplify the inner expression or square roots. With practice and patience, you can become proficient in simplifying exponents and roots.