The Expression $\sqrt[4]{3^5} \cdot \sqrt[3]{3^4}$ Is Equivalent To:A. $3^{\frac{5}{3}}$ B. $9^{\frac{31}{12}}$ C. $3^{\frac{31}{2}}$ D. $9^{\frac{5}{3}}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. One of the most common ways to simplify expressions is by using the properties of exponents. In this article, we will explore the expression 354⋅343\sqrt[4]{3^5} \cdot \sqrt[3]{3^4} and simplify it using the properties of exponents.

Understanding the Expression

The given expression is a product of two radical expressions. The first radical expression is 354\sqrt[4]{3^5}, which can be rewritten as (35)14(3^5)^{\frac{1}{4}}. The second radical expression is 343\sqrt[3]{3^4}, which can be rewritten as (34)13(3^4)^{\frac{1}{3}}.

Simplifying the Expression

To simplify the expression, we can use the property of exponents that states (am)n=amn(a^m)^n = a^{mn}. Applying this property to the first radical expression, we get:

354=(35)14=354\sqrt[4]{3^5} = (3^5)^{\frac{1}{4}} = 3^{\frac{5}{4}}

Similarly, applying the property to the second radical expression, we get:

343=(34)13=343\sqrt[3]{3^4} = (3^4)^{\frac{1}{3}} = 3^{\frac{4}{3}}

Now, we can multiply the two simplified expressions together:

354⋅343=354⋅343\sqrt[4]{3^5} \cdot \sqrt[3]{3^4} = 3^{\frac{5}{4}} \cdot 3^{\frac{4}{3}}

Using the Product of Powers Property

The product of powers property states that am⋅an=am+na^m \cdot a^n = a^{m+n}. We can use this property to simplify the expression further:

354⋅343=354+433^{\frac{5}{4}} \cdot 3^{\frac{4}{3}} = 3^{\frac{5}{4} + \frac{4}{3}}

To add the fractions, we need to find a common denominator. The least common multiple of 4 and 3 is 12. So, we can rewrite the fractions with a common denominator:

54=1512\frac{5}{4} = \frac{15}{12}

43=1612\frac{4}{3} = \frac{16}{12}

Now, we can add the fractions:

1512+1612=3112\frac{15}{12} + \frac{16}{12} = \frac{31}{12}

So, the simplified expression is:

354+43=331123^{\frac{5}{4} + \frac{4}{3}} = 3^{\frac{31}{12}}

Conclusion

In this article, we simplified the expression 354⋅343\sqrt[4]{3^5} \cdot \sqrt[3]{3^4} using the properties of exponents. We first rewrote the radical expressions using the property of exponents, then multiplied the simplified expressions together, and finally used the product of powers property to simplify the expression further. The final simplified expression is 331123^{\frac{31}{12}}.

Comparison with Answer Choices

Now, let's compare our simplified expression with the answer choices:

  • A. 3533^{\frac{5}{3}}
  • B. 931129^{\frac{31}{12}}
  • C. 33123^{\frac{31}{2}}
  • D. 9539^{\frac{5}{3}}

Our simplified expression is 331123^{\frac{31}{12}}, which is not equal to any of the answer choices. However, we can rewrite the expression as 931129^{\frac{31}{12}} by using the property of exponents that states am=(an)mna^m = (a^n)^{\frac{m}{n}}. Since 9=329 = 3^2, we can rewrite the expression as:

33112=(32)3112=931123^{\frac{31}{12}} = (3^2)^{\frac{31}{12}} = 9^{\frac{31}{12}}

Therefore, the correct answer is B. 931129^{\frac{31}{12}}.

Final Answer

Introduction

In our previous article, we simplified the expression 354⋅343\sqrt[4]{3^5} \cdot \sqrt[3]{3^4} using the properties of exponents. In this article, we will answer some frequently asked questions related to the expression and provide additional insights.

Q: What is the property of exponents used to simplify the expression?

A: The property of exponents used to simplify the expression is (am)n=amn(a^m)^n = a^{mn}. This property allows us to rewrite the radical expressions as powers of 3.

Q: How do you rewrite the radical expressions using the property of exponents?

A: To rewrite the radical expressions, we can use the property of exponents to rewrite the expressions as powers of 3. For example, 354\sqrt[4]{3^5} can be rewritten as (35)14=354(3^5)^{\frac{1}{4}} = 3^{\frac{5}{4}}.

Q: What is the product of powers property?

A: The product of powers property states that am⋅an=am+na^m \cdot a^n = a^{m+n}. This property allows us to simplify the expression by adding the exponents.

Q: How do you add the exponents using the product of powers property?

A: To add the exponents, we need to find a common denominator. The least common multiple of 4 and 3 is 12. So, we can rewrite the fractions with a common denominator:

54=1512\frac{5}{4} = \frac{15}{12}

43=1612\frac{4}{3} = \frac{16}{12}

Now, we can add the fractions:

1512+1612=3112\frac{15}{12} + \frac{16}{12} = \frac{31}{12}

Q: What is the final simplified expression?

A: The final simplified expression is 331123^{\frac{31}{12}}.

Q: Can the expression be rewritten in a different form?

A: Yes, the expression can be rewritten in a different form. Since 9=329 = 3^2, we can rewrite the expression as:

33112=(32)3112=931123^{\frac{31}{12}} = (3^2)^{\frac{31}{12}} = 9^{\frac{31}{12}}

Q: What is the correct answer choice?

A: The correct answer choice is B. 931129^{\frac{31}{12}}.

Conclusion

In this article, we answered some frequently asked questions related to the expression 354⋅343\sqrt[4]{3^5} \cdot \sqrt[3]{3^4} and provided additional insights. We hope this article has been helpful in understanding the properties of exponents and how to simplify expressions.

Additional Resources

For more information on the properties of exponents and how to simplify expressions, please refer to the following resources:

  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions

Final Answer

The final answer is B. 931129^{\frac{31}{12}}.