Determine Which Expressions Are Equivalent To $7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$.\begin{tabular}{|c|c|}\hlineExpression & Equivalent Or Not Equivalent \\hline$343$ & ___ \\hline$ 49 49 49 [/tex] & ___

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Determine which expressions are equivalent to $7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$

Understanding the Problem

The given problem involves determining which expressions are equivalent to the expression $7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$. To solve this problem, we need to simplify the given expression and then compare it with the given options.

Simplifying the Expression

We can simplify the expression $7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$ by using the properties of exponents. We know that $49 = 7^2$, so we can rewrite the expression as:

715⋅(72)757^{\frac{1}{5}} \cdot (7^2)^{\frac{7}{5}}

Using the property of exponents that states $(am)n = a^{mn}$, we can simplify the expression further:

715⋅71457^{\frac{1}{5}} \cdot 7^{\frac{14}{5}}

Now, we can combine the two terms using the property of exponents that states $a^m \cdot a^n = a^{m+n}$:

715+1457^{\frac{1}{5} + \frac{14}{5}}

Simplifying the exponent, we get:

71557^{\frac{15}{5}}

737^3

Evaluating the Options

Now that we have simplified the expression, we can compare it with the given options.

Option 1: 343

We know that $343 = 7^3$, so this option is equivalent to the simplified expression.

Option 2: 49

We know that $49 = 7^2$, so this option is not equivalent to the simplified expression.

Conclusion

Based on our analysis, we can conclude that the expression $7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$ is equivalent to $343$, but not equivalent to $49$.

Key Takeaways

  • To simplify the expression $7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}}$, we can use the properties of exponents.
  • We can rewrite $49$ as $7^2$ and then simplify the expression further.
  • The simplified expression is $7^3$, which is equivalent to $343$.

Additional Practice Problems

  • Simplify the expression $2^{\frac{1}{3}} \cdot 8^{\frac{5}{3}}$.
  • Determine which expressions are equivalent to $2^{\frac{1}{4}} \cdot 16^{\frac{3}{4}}$.

Answer Key

  • The simplified expression is $2^2$, which is equivalent to $4$.
  • The expressions $4$ and $64$ are equivalent to $2^{\frac{1}{4}} \cdot 16^{\frac{3}{4}}$.

Discussion

This problem requires the use of properties of exponents to simplify the given expression. The student needs to understand the concept of exponents and how to apply them to simplify expressions. The problem also requires the student to evaluate the options and determine which one is equivalent to the simplified expression.

Mathematical Concepts

  • Properties of exponents
  • Simplifying expressions using exponents
  • Evaluating expressions

Difficulty Level

  • This problem is considered to be of moderate difficulty, as it requires the student to apply the properties of exponents to simplify the expression and then evaluate the options.

Time Required

  • This problem should take approximately 15-20 minutes to complete, depending on the student's level of understanding and familiarity with the concepts.

Assessment

  • This problem can be used as a formative assessment to evaluate the student's understanding of the properties of exponents and their ability to simplify expressions.

Extension

  • This problem can be extended by asking the student to simplify more complex expressions using the properties of exponents.

Real-World Applications

  • This problem has real-world applications in fields such as engineering, physics, and computer science, where the use of exponents is common.

Conclusion

In conclusion, this problem requires the student to apply the properties of exponents to simplify the given expression and then evaluate the options. The student needs to understand the concept of exponents and how to apply them to simplify expressions. The problem also requires the student to evaluate the options and determine which one is equivalent to the simplified expression.
Q&A: Determining Equivalent Expressions

Frequently Asked Questions

Q: What is the main concept behind determining equivalent expressions?

A: The main concept behind determining equivalent expressions is to simplify the given expression using the properties of exponents and then compare it with the given options.

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

A: To simplify an expression using the properties of exponents, you need to apply the following rules:

  • (am)n=amn(a^m)^n = a^{mn}

  • am⋅an=am+na^m \cdot a^n = a^{m+n}

  • am÷an=am−na^m \div a^n = a^{m-n}

Q: What is the difference between equivalent and non-equivalent expressions?

A: Equivalent expressions are expressions that have the same value, while non-equivalent expressions are expressions that have different values.

Q: How do I determine which expressions are equivalent to a given expression?

A: To determine which expressions are equivalent to a given expression, you need to simplify the given expression using the properties of exponents and then compare it with the given options.

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

A: Some common mistakes to avoid when determining equivalent expressions include:

  • Not simplifying the expression using the properties of exponents
  • Not comparing the simplified expression with the given options
  • Not considering the order of operations when simplifying the expression

Q: How can I practice determining equivalent expressions?

A: You can practice determining equivalent expressions by working on problems that involve simplifying expressions using the properties of exponents and then comparing the simplified expression with the given options.

Q: What are some real-world applications of determining equivalent expressions?

A: Some real-world applications of determining equivalent expressions include:

  • Engineering: Determining equivalent expressions is used in engineering to simplify complex mathematical expressions and solve problems.
  • Physics: Determining equivalent expressions is used in physics to simplify complex mathematical expressions and solve problems.
  • Computer Science: Determining equivalent expressions is used in computer science to simplify complex mathematical expressions and solve problems.

Q: How can I use technology to help me determine equivalent expressions?

A: You can use technology such as calculators and computer software to help you determine equivalent expressions. These tools can simplify complex mathematical expressions and provide you with the equivalent expression.

Q: What are some common errors to watch out for when determining equivalent expressions?

A: Some common errors to watch out for when determining equivalent expressions include:

  • Not simplifying the expression using the properties of exponents
  • Not comparing the simplified expression with the given options
  • Not considering the order of operations when simplifying the expression

Q: How can I check my work when determining equivalent expressions?

A: You can check your work by:

  • Simplifying the expression using the properties of exponents
  • Comparing the simplified expression with the given options
  • Considering the order of operations when simplifying the expression

Conclusion

Determining equivalent expressions is an important concept in mathematics that requires the use of properties of exponents to simplify complex mathematical expressions. By understanding the concept of equivalent expressions and practicing determining equivalent expressions, you can improve your problem-solving skills and apply them to real-world problems.

Additional Resources

  • Khan Academy: Determining Equivalent Expressions
  • Mathway: Determining Equivalent Expressions
  • Wolfram Alpha: Determining Equivalent Expressions

Practice Problems

  • Simplify the expression $2^{\frac{1}{3}} \cdot 8^{\frac{5}{3}}$ and determine which expressions are equivalent to it.
  • Simplify the expression $3^{\frac{1}{2}} \cdot 9^{\frac{3}{2}}$ and determine which expressions are equivalent to it.

Answer Key

  • The simplified expression is $2^2$, which is equivalent to $4$.
  • The simplified expression is $3^3$, which is equivalent to $27$.