Select The Simplification That Accurately Explains The Following Statement: 7 3 = 7 1 3 \sqrt[3]{7} = 7^{\frac{1}{3}} 3 7 ​ = 7 3 1 ​ A. $\left(7 {\frac{1}{3}}\right) 3 = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7^{\frac{1}{3} \cdot \frac{1}{3}

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Introduction

In mathematics, radicals and exponents are two ways to represent the same mathematical operation. While radicals are often used to represent roots, exponents are used to represent repeated multiplication. In this article, we will explore the relationship between radicals and exponents, specifically the statement 73=713\sqrt[3]{7} = 7^{\frac{1}{3}}. We will examine the properties of radicals and exponents, and use mathematical reasoning to determine the simplification that accurately explains this statement.

Radicals and Exponents: A Brief Overview

Radicals are a way to represent roots of numbers. For example, 7\sqrt{7} represents the square root of 7, which is a number that, when multiplied by itself, gives 7. Similarly, 73\sqrt[3]{7} represents the cube root of 7, which is a number that, when multiplied by itself twice, gives 7.

Exponents, on the other hand, are a way to represent repeated multiplication. For example, 727^2 represents 7 multiplied by itself, which is equal to 49. Similarly, 737^3 represents 7 multiplied by itself three times, which is equal to 343.

The Relationship Between Radicals and Exponents

The statement 73=713\sqrt[3]{7} = 7^{\frac{1}{3}} suggests that there is a relationship between radicals and exponents. To understand this relationship, let's consider the properties of radicals and exponents.

Property 1: Radicals can be rewritten as exponents

We know that 73\sqrt[3]{7} represents the cube root of 7. This can be rewritten as 7137^{\frac{1}{3}}, which represents 7 multiplied by itself one-third of a time.

Property 2: Exponents can be rewritten as radicals

We know that 7137^{\frac{1}{3}} represents 7 multiplied by itself one-third of a time. This can be rewritten as 73\sqrt[3]{7}, which represents the cube root of 7.

Property 3: Radicals and exponents are equivalent

Based on properties 1 and 2, we can conclude that radicals and exponents are equivalent. This means that 73\sqrt[3]{7} and 7137^{\frac{1}{3}} represent the same mathematical operation.

Simplifying the Statement

Now that we have established the relationship between radicals and exponents, let's simplify the statement 73=713\sqrt[3]{7} = 7^{\frac{1}{3}}.

Option A: (713)3=713⋅713⋅713=713⋅13\left(7^{\frac{1}{3}}\right)^3 = 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} = 7^{\frac{1}{3} \cdot \frac{1}{3}}

This option suggests that raising 7137^{\frac{1}{3}} to the power of 3 is equivalent to multiplying 7137^{\frac{1}{3}} by itself three times. This is a correct statement, as raising a number to a power is equivalent to multiplying that number by itself that many times.

Option B: (713)3=713⋅3=71=7\left(7^{\frac{1}{3}}\right)^3 = 7^{\frac{1}{3} \cdot 3} = 7^1 = 7

This option suggests that raising 7137^{\frac{1}{3}} to the power of 3 is equivalent to multiplying 7137^{\frac{1}{3}} by itself three times, and then simplifying the result to 7. This is also a correct statement, as raising a number to a power is equivalent to multiplying that number by itself that many times.

Conclusion

In conclusion, the statement 73=713\sqrt[3]{7} = 7^{\frac{1}{3}} accurately represents the relationship between radicals and exponents. We have established that radicals and exponents are equivalent, and have simplified the statement using mathematical reasoning. The correct simplification is (713)3=713⋅13\left(7^{\frac{1}{3}}\right)^3 = 7^{\frac{1}{3} \cdot \frac{1}{3}}, which represents the cube root of 7 multiplied by itself three times.

References

  • [1] "Radicals and Exponents" by Math Open Reference
  • [2] "Exponents and Radicals" by Khan Academy
  • [3] "Radicals and Exponents" by Purplemath

Additional Resources

  • [1] "Radicals and Exponents" by Mathway
  • [2] "Exponents and Radicals" by IXL
  • [3] "Radicals and Exponents" by Wolfram Alpha
    Frequently Asked Questions: Radicals and Exponents =====================================================

Q: What is the difference between radicals and exponents?

A: Radicals and exponents are two ways to represent the same mathematical operation. Radicals are used to represent roots, while exponents are used to represent repeated multiplication.

Q: How do I convert a radical to an exponent?

A: To convert a radical to an exponent, you can use the following formula:

xn=x1n\sqrt[n]{x} = x^{\frac{1}{n}}

For example, 73=713\sqrt[3]{7} = 7^{\frac{1}{3}}.

Q: How do I convert an exponent to a radical?

A: To convert an exponent to a radical, you can use the following formula:

x1n=xnx^{\frac{1}{n}} = \sqrt[n]{x}

For example, 713=737^{\frac{1}{3}} = \sqrt[3]{7}.

Q: What is the relationship between radicals and exponents?

A: Radicals and exponents are equivalent. This means that xn\sqrt[n]{x} and x1nx^{\frac{1}{n}} represent the same mathematical operation.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the following steps:

  1. Look for any perfect squares or cubes in the radicand (the number inside the radical).
  2. If there are any perfect squares or cubes, simplify the radicand by taking the square root or cube root of the perfect square or cube.
  3. If there are no perfect squares or cubes, the radical expression cannot be simplified further.

Q: How do I simplify an exponent expression?

A: To simplify an exponent expression, you can use the following steps:

  1. Look for any common factors in the base (the number being raised to a power) and the exponent.
  2. If there are any common factors, simplify the expression by canceling out the common factors.
  3. If there are no common factors, the exponent expression cannot be simplified further.

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

A: The order of operations for radicals and exponents is the same as the order of operations for regular arithmetic:

  1. Parentheses: Evaluate any expressions inside parentheses first.
  2. Exponents: Evaluate any exponents next.
  3. Radicals: Evaluate any radicals next.
  4. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  5. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Can I use a calculator to evaluate radical expressions?

A: Yes, you can use a calculator to evaluate radical expressions. However, keep in mind that calculators may not always display the exact value of a radical expression. In some cases, the calculator may display an approximate value or a decimal approximation.

Q: Can I use a calculator to evaluate exponent expressions?

A: Yes, you can use a calculator to evaluate exponent expressions. However, keep in mind that calculators may not always display the exact value of an exponent expression. In some cases, the calculator may display an approximate value or a decimal approximation.

Conclusion

In conclusion, radicals and exponents are two ways to represent the same mathematical operation. By understanding the relationship between radicals and exponents, you can simplify complex expressions and evaluate them more easily. Remember to follow the order of operations and use a calculator only when necessary.

References

  • [1] "Radicals and Exponents" by Math Open Reference
  • [2] "Exponents and Radicals" by Khan Academy
  • [3] "Radicals and Exponents" by Purplemath

Additional Resources

  • [1] "Radicals and Exponents" by Mathway
  • [2] "Exponents and Radicals" by IXL
  • [3] "Radicals and Exponents" by Wolfram Alpha