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

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Which of these choices show a pair of equivalent expressions? Check all that apply

Understanding Equivalent Expressions

Equivalent expressions are mathematical expressions that have the same value, even if they are written differently. In other words, they are expressions that can be simplified to the same result. In this article, we will explore four different pairs of expressions and determine which ones are equivalent.

A. 52/35^{2 / 3} and (53)2(\sqrt[3]{5})^2

To determine if these two expressions are equivalent, we need to simplify them. The first expression, 52/35^{2 / 3}, can be rewritten as (51/3)2(5^{1/3})^2. This is because when we have a power raised to another power, we multiply the exponents. So, 52/35^{2 / 3} is equivalent to (51/3)2(5^{1/3})^2.

The second expression, (53)2(\sqrt[3]{5})^2, can be rewritten as 52/35^{2/3}. This is because the cube root of a number can be written as the number raised to the power of 1/3. So, (53)2(\sqrt[3]{5})^2 is equivalent to 52/35^{2/3}.

Since both expressions are equivalent to 52/35^{2/3}, we can conclude that A. 52/35^{2 / 3} and (53)2(\sqrt[3]{5})^2 are equivalent expressions.

B. (814)5(\sqrt[4]{81})^5 and 815/481^{5 / 4}

To determine if these two expressions are equivalent, we need to simplify them. The first expression, (814)5(\sqrt[4]{81})^5, can be rewritten as 815/481^{5/4}. This is because the fourth root of a number can be written as the number raised to the power of 1/4. So, (814)5(\sqrt[4]{81})^5 is equivalent to 815/481^{5/4}.

The second expression, 815/481^{5 / 4}, is already in its simplest form. However, we can rewrite it as (811/4)5(81^{1/4})^5. This is because when we have a power raised to another power, we multiply the exponents. So, 815/481^{5 / 4} is equivalent to (811/4)5(81^{1/4})^5.

Since both expressions are equivalent to 815/481^{5/4}, we can conclude that B. (814)5(\sqrt[4]{81})^5 and 815/481^{5 / 4} are equivalent expressions.

C. 75/77^{5 / 7} and (77)5(\sqrt[7]{7})^5

To determine if these two expressions are equivalent, we need to simplify them. The first expression, 75/77^{5 / 7}, can be rewritten as (71/7)5(7^{1/7})^5. This is because when we have a power raised to another power, we multiply the exponents. So, 75/77^{5 / 7} is equivalent to (71/7)5(7^{1/7})^5.

The second expression, (77)5(\sqrt[7]{7})^5, can be rewritten as 75/77^{5/7}. This is because the seventh root of a number can be written as the number raised to the power of 1/7. So, (77)5(\sqrt[7]{7})^5 is equivalent to 75/77^{5/7}.

Since both expressions are equivalent to 75/77^{5/7}, we can conclude that C. 75/77^{5 / 7} and (77)5(\sqrt[7]{7})^5 are equivalent expressions.

D.

Unfortunately, there is no expression D to evaluate.

Conclusion

In conclusion, we have determined that the following pairs of expressions are equivalent:

  • A. 52/35^{2 / 3} and (53)2(\sqrt[3]{5})^2
  • B. (814)5(\sqrt[4]{81})^5 and 815/481^{5 / 4}
  • C. 75/77^{5 / 7} and (77)5(\sqrt[7]{7})^5

These expressions are equivalent because they can be simplified to the same result. Understanding equivalent expressions is an important concept in mathematics, as it allows us to simplify complex expressions and solve problems more easily.

References

Q: What are equivalent expressions?

A: Equivalent expressions are mathematical expressions that have the same value, even if they are written differently. In other words, they are expressions that can be simplified to the same result.

Q: Why are equivalent expressions important?

A: Equivalent expressions are important because they allow us to simplify complex expressions and solve problems more easily. By recognizing equivalent expressions, we can rewrite them in a simpler form, making it easier to work with them.

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

A: To determine if two expressions are equivalent, you need to simplify them and see if they have the same value. You can use various techniques, such as factoring, canceling out common factors, or using the properties of exponents.

Q: What are some common techniques for simplifying expressions?

A: Some common techniques for simplifying expressions include:

  • Factoring: breaking down an expression into simpler factors
  • Canceling out common factors: canceling out common factors in the numerator and denominator of a fraction
  • Using the properties of exponents: using the rules of exponents to simplify expressions

Q: Can you give an example of how to simplify an expression using the properties of exponents?

A: Yes, here's an example:

Suppose we have the expression 23⋅242^3 \cdot 2^4. To simplify this expression, we can use the property of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}. Applying this property, we get:

23⋅24=23+4=272^3 \cdot 2^4 = 2^{3+4} = 2^7

Q: How do I know if an expression is in its simplest form?

A: An expression is in its simplest form if it cannot be simplified further using the techniques mentioned above. In other words, if you have simplified an expression as much as possible, and it cannot be simplified further, then it is in its simplest form.

Q: Can you give an example of an expression that is not in its simplest form?

A: Yes, here's an example:

Suppose we have the expression 6⋅86 \cdot 8. This expression is not in its simplest form because we can simplify it further by multiplying the numbers together:

6⋅8=486 \cdot 8 = 48

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

A: To determine if two expressions are equivalent, you need to simplify them and see if they have the same value. You can use various techniques, such as factoring, canceling out common factors, or using the properties of exponents.

Q: Can you give an example of how to determine if two expressions are equivalent?

A: Yes, here's an example:

Suppose we have the expressions x2+4x+4x^2 + 4x + 4 and (x+2)2(x + 2)^2. To determine if these expressions are equivalent, we can simplify them and see if they have the same value:

x2+4x+4=(x+2)2x^2 + 4x + 4 = (x + 2)^2

Since both expressions have the same value, we can conclude that they are equivalent.

Q: What are some common mistakes to avoid when working with equivalent expressions?

A: Some common mistakes to avoid when working with equivalent expressions include:

  • Not simplifying expressions enough
  • Not recognizing equivalent expressions
  • Not using the properties of exponents correctly

Q: Can you give an example of a common mistake to avoid?

A: Yes, here's an example:

Suppose we have the expression 23⋅242^3 \cdot 2^4. If we simplify this expression incorrectly, we might get:

23⋅24=27⋅232^3 \cdot 2^4 = 2^7 \cdot 2^3

This is incorrect because we have not used the property of exponents correctly. The correct simplification is:

23⋅24=23+4=272^3 \cdot 2^4 = 2^{3+4} = 2^7

Q: How can I practice working with equivalent expressions?

A: You can practice working with equivalent expressions by:

  • Simplifying expressions and checking if they are equivalent
  • Using the properties of exponents to simplify expressions
  • Working with different types of expressions, such as linear and quadratic expressions

Q: What are some resources for learning more about equivalent expressions?

A: Some resources for learning more about equivalent expressions include:

  • Textbooks and online resources that cover algebra and mathematics
  • Online tutorials and videos that explain equivalent expressions
  • Practice problems and worksheets that allow you to practice working with equivalent expressions