Which Of These Choices Show A Pair Of Equivalent Expressions? Check All That Apply.A. $(\sqrt[4]{81})^5$ And $81^{5/4}$B. $7^{5/7}$ And $(\sqrt{7})^5$C. $5^{2/3}$ And $(\sqrt{5})^3$D.

Introduction

In mathematics, equivalent expressions are those that have the same value, even if they are written differently. Identifying equivalent expressions is a crucial skill in algebra and other branches of mathematics. In this article, we will explore the concept of equivalent expressions and examine the given choices to determine which pairs are equivalent.

Understanding Equivalent Expressions

Equivalent expressions are expressions that have the same value, but may be written in different forms. For example, the expressions $2 \times 3$ and $6$ are equivalent because they both represent the same value. Similarly, the expressions $\sqrt{4}$ and $2$ are equivalent because they both represent the same value.

Properties of Exponents

Exponents are a fundamental concept in mathematics, and understanding their properties is essential for identifying equivalent expressions. The properties of exponents include:

• Product of Powers Property: When multiplying two powers with the same base, add the exponents. For example, $a^m \times a^n = a^{m+n}$.
• Power of a Power Property: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \times n}$.
• Quotient of Powers Property: When dividing two powers with the same base, subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Analyzing the Choices

Now that we have a solid understanding of equivalent expressions and the properties of exponents, let's analyze the given choices.

A. $(\sqrt[4]{81})^5$ and $81^{5/4}$

To determine if these expressions are equivalent, we need to simplify them using the properties of exponents.

• Simplifying $(\sqrt[4]{81})^5$: Using the property of exponents, we can rewrite this expression as $(81^{1/4})^5 = 81^{5/4}$.
• Simplifying $81^{5/4}$: This expression is already simplified.

Since both expressions simplify to the same value, they are equivalent.

B. $7^{5/7}$ and $(\sqrt{7})^5$

To determine if these expressions are equivalent, we need to simplify them using the properties of exponents.

• Simplifying $7^{5/7}$: Using the property of exponents, we can rewrite this expression as $(7^{1/7})^5 = 7^{5/7}$.
• Simplifying $(\sqrt{7})^5$: Using the property of exponents, we can rewrite this expression as $(7^{1/2})^5 = 7^{5/2}$.

Since these expressions do not simplify to the same value, they are not equivalent.

C. $5^{2/3}$ and $(\sqrt{5})^3$

To determine if these expressions are equivalent, we need to simplify them using the properties of exponents.

• Simplifying $5^{2/3}$: Using the property of exponents, we can rewrite this expression as $(5^{1/3})^2 = 5^{2/3}$.
• Simplifying $(\sqrt{5})^3$: Using the property of exponents, we can rewrite this expression as $(5^{1/2})^3 = 5^{3/2}$.

Since these expressions do not simplify to the same value, they are not equivalent.

D.

There is no expression provided for choice D.

Conclusion

In conclusion, the only pair of equivalent expressions among the given choices is A. $(\sqrt[4]{81})^5$ and $81^{5/4}$. These expressions simplify to the same value using the properties of exponents.

Key Takeaways

• Equivalent expressions are those that have the same value, even if they are written differently.
• Understanding the properties of exponents is essential for identifying equivalent expressions.
• The product of powers property, power of a power property, and quotient of powers property are fundamental concepts in mathematics that can be used to simplify expressions and identify equivalent expressions.

Final Thoughts

Frequently Asked Questions

Q: What are equivalent expressions?

A: Equivalent expressions are expressions that have the same value, even if they are written differently.

Q: Why are equivalent expressions important in mathematics?

A: Equivalent expressions are important in mathematics because they allow us to simplify complex expressions and solve problems more efficiently. By identifying equivalent expressions, we can rewrite them in a more convenient form, making it easier to perform calculations and arrive at the correct solution.

Q: How do I identify equivalent expressions?

A: To identify equivalent expressions, you need to understand the properties of exponents, including the product of powers property, power of a power property, and quotient of powers property. By applying these properties, you can simplify expressions and determine if they are equivalent.

Q: What are some common examples of equivalent expressions?

A: Some common examples of equivalent expressions include:

• $2 \times 3$ and $6$
• $\sqrt{4}$ and $2$
• $(\sqrt[4]{81})^5$ and $81^{5/4}$

Q: Can you provide more examples of equivalent expressions?

A: Here are some more examples of equivalent expressions:

• $3^2$ and $(3^1)^2$
• $4^3$ and $(4^1)^3$
• $2^5$ and $(2^1)^5$

Q: How do I simplify expressions using the properties of exponents?

A: To simplify expressions using the properties of exponents, follow these steps:

1. Identify the properties of exponents that apply to the expression.
2. Apply the properties of exponents to simplify the expression.
3. Check if the simplified expression is equivalent to the original expression.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Forgetting to apply the properties of exponents
• Making errors when simplifying expressions
• Not checking if the simplified expression is equivalent to the original expression

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, follow these steps:

1. Simplify both expressions using the properties of exponents.
2. Check if the simplified expressions are the same.
3. If the simplified expressions are the same, then the original expressions are equivalent.

Q: Can you provide a step-by-step example of how to determine if two expressions are equivalent?

A: Here is a step-by-step example of how to determine if two expressions are equivalent:

• Step 1: Simplify both expressions using the properties of exponents.
• Step 2: Check if the simplified expressions are the same.
• Step 3: If the simplified expressions are the same, then the original expressions are equivalent.

Conclusion

In conclusion, equivalent expressions are a fundamental concept in mathematics that allows us to simplify complex expressions and solve problems more efficiently. By understanding the properties of exponents and applying them to simplify expressions, we can determine if two expressions are equivalent. Remember to avoid common mistakes and follow the steps outlined above to determine if two expressions are equivalent.

Key Takeaways

• Equivalent expressions are expressions that have the same value, even if they are written differently.
• Understanding the properties of exponents is essential for identifying equivalent expressions.
• The product of powers property, power of a power property, and quotient of powers property are fundamental concepts in mathematics that can be used to simplify expressions and identify equivalent expressions.

Final Thoughts

Identifying equivalent expressions is a crucial skill in mathematics that requires a solid understanding of the properties of exponents. By applying the properties of exponents and simplifying expressions, we can determine which expressions are equivalent and which are not. Remember to avoid common mistakes and follow the steps outlined above to determine if two expressions are equivalent.