What Is $\frac{\sqrt[3]{3}}{\sqrt[3]{2x}}$ In Simplest Form?A. $\frac{\sqrt[3]{12x^2}}{2x}$ B. $ 6 X 3 2 X \frac{\sqrt[3]{6x}}{2x} 2 X 3 6 X ​ ​ [/tex] C. $\frac{\sqrt[3]{3}}{2x}$

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Understanding the Problem

To simplify the given expression $\frac{\sqrt[3]{3}}{\sqrt[3]{2x}}$, we need to apply the rules of exponents and radicals. The expression involves cube roots, which can be simplified using the properties of radicals.

Simplifying the Cube Roots

The cube root of a number can be expressed as a fractional exponent. For example, $\sqrt[3]{a} = a^{\frac{1}{3}}$. Using this property, we can rewrite the given expression as:

332x3=313(2x)13\frac{\sqrt[3]{3}}{\sqrt[3]{2x}} = \frac{3^{\frac{1}{3}}}{(2x)^{\frac{1}{3}}}

Applying the Quotient Rule for Exponents

When dividing two powers with the same base, we subtract the exponents. In this case, we have:

313(2x)13=313−13⋅(2x)−13\frac{3^{\frac{1}{3}}}{(2x)^{\frac{1}{3}}} = 3^{\frac{1}{3} - \frac{1}{3}} \cdot (2x)^{-\frac{1}{3}}

Simplifying the Expression

Now, we can simplify the expression further by evaluating the exponents:

313−13⋅(2x)−13=30⋅(2x)−133^{\frac{1}{3} - \frac{1}{3}} \cdot (2x)^{-\frac{1}{3}} = 3^0 \cdot (2x)^{-\frac{1}{3}}

Evaluating the Exponents

Since any number raised to the power of 0 is equal to 1, we have:

30⋅(2x)−13=1⋅(2x)−133^0 \cdot (2x)^{-\frac{1}{3}} = 1 \cdot (2x)^{-\frac{1}{3}}

Simplifying the Radical

Now, we can simplify the radical by applying the property of negative exponents:

(2x)−13=1(2x)13(2x)^{-\frac{1}{3}} = \frac{1}{(2x)^{\frac{1}{3}}}

Rewriting the Expression

Using the property of radicals, we can rewrite the expression as:

1(2x)13=12x3\frac{1}{(2x)^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{2x}}

Simplifying the Expression

Now, we can simplify the expression further by rewriting it as a fraction:

12x3=123⋅x3\frac{1}{\sqrt[3]{2x}} = \frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{x}}

Simplifying the Cube Roots

Using the property of cube roots, we can rewrite the expression as:

123⋅x3=123⋅x13\frac{1}{\sqrt[3]{2} \cdot \sqrt[3]{x}} = \frac{1}{\sqrt[3]{2} \cdot x^{\frac{1}{3}}}

Simplifying the Expression

Now, we can simplify the expression further by rewriting it as a fraction:

123⋅x13=123⋅1x13\frac{1}{\sqrt[3]{2} \cdot x^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{2}} \cdot \frac{1}{x^{\frac{1}{3}}}

Simplifying the Cube Roots

Using the property of cube roots, we can rewrite the expression as:

123⋅1x13=123⋅1x3\frac{1}{\sqrt[3]{2}} \cdot \frac{1}{x^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{2}} \cdot \frac{1}{\sqrt[3]{x}}

Simplifying the Expression

Now, we can simplify the expression further by rewriting it as a fraction:

123⋅1x3=12x3\frac{1}{\sqrt[3]{2}} \cdot \frac{1}{\sqrt[3]{x}} = \frac{1}{\sqrt[3]{2x}}

Conclusion

The simplified form of the given expression $\frac{\sqrt[3]{3}}{\sqrt[3]{2x}}$ is:

332x3=3323⋅x3=3323⋅1x3=3323⋅1x3=332x3\frac{\sqrt[3]{3}}{\sqrt[3]{2x}} = \frac{\sqrt[3]{3}}{\sqrt[3]{2} \cdot \sqrt[3]{x}} = \frac{\sqrt[3]{3}}{\sqrt[3]{2}} \cdot \frac{1}{\sqrt[3]{x}} = \frac{\sqrt[3]{3}}{\sqrt[3]{2}} \cdot \frac{1}{\sqrt[3]{x}} = \frac{\sqrt[3]{3}}{\sqrt[3]{2x}}

Final Answer

The final answer is: 332x\boxed{\frac{\sqrt[3]{3}}{2x}}

Q: What is the simplified form of $\frac{\sqrt[3]{3}}{\sqrt[3]{2x}}$?

A: The simplified form of $\frac{\sqrt[3]{3}}{\sqrt[3]{2x}}$ is $\frac{\sqrt[3]{3}}{2x}$.

Q: How do I simplify an expression with cube roots?

A: To simplify an expression with cube roots, you can use the properties of radicals and exponents. You can rewrite the cube root as a fractional exponent and then apply the quotient rule for exponents.

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that when dividing two powers with the same base, you subtract the exponents. For example, $\frac{am}{an} = a^{m-n}$.

Q: How do I simplify a radical with a negative exponent?

A: To simplify a radical with a negative exponent, you can rewrite it as a fraction. For example, $(2x)^{-\frac{1}{3}} = \frac{1}{(2x)^{\frac{1}{3}}}$.

Q: What is the property of cube roots?

A: The property of cube roots states that $\sqrt[3]{a} = a^{\frac{1}{3}}$.

Q: How do I simplify an expression with multiple cube roots?

A: To simplify an expression with multiple cube roots, you can use the properties of radicals and exponents. You can rewrite each cube root as a fractional exponent and then apply the quotient rule for exponents.

Q: What is the final answer to the expression $\frac{\sqrt[3]{3}}{\sqrt[3]{2x}}$?

A: The final answer to the expression $\frac{\sqrt[3]{3}}{\sqrt[3]{2x}}$ is $\frac{\sqrt[3]{3}}{2x}$.

Q: Can you provide more examples of simplifying expressions with cube roots?

A: Yes, here are a few more examples:

  • 43x3=43x3\frac{\sqrt[3]{4}}{\sqrt[3]{x}} = \frac{\sqrt[3]{4}}{\sqrt[3]{x}}

  • 933x3=9333⋅x3=9333⋅1x3=9333⋅1x3\frac{\sqrt[3]{9}}{\sqrt[3]{3x}} = \frac{\sqrt[3]{9}}{\sqrt[3]{3} \cdot \sqrt[3]{x}} = \frac{\sqrt[3]{9}}{\sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{x}} = \frac{\sqrt[3]{9}}{\sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{x}}

  • 1634x3=16343⋅x3=16343⋅1x3=16343⋅1x3\frac{\sqrt[3]{16}}{\sqrt[3]{4x}} = \frac{\sqrt[3]{16}}{\sqrt[3]{4} \cdot \sqrt[3]{x}} = \frac{\sqrt[3]{16}}{\sqrt[3]{4}} \cdot \frac{1}{\sqrt[3]{x}} = \frac{\sqrt[3]{16}}{\sqrt[3]{4}} \cdot \frac{1}{\sqrt[3]{x}}

Q: How do I know when to use the quotient rule for exponents?

A: You should use the quotient rule for exponents when dividing two powers with the same base. This rule allows you to simplify expressions with exponents and radicals.

Q: Can you provide more information on the properties of radicals and exponents?

A: Yes, here are a few more properties of radicals and exponents:

  • an=a1n\sqrt[n]{a} = a^{\frac{1}{n}}

  • amn=(am)1n=amn\sqrt[n]{a^m} = (a^m)^{\frac{1}{n}} = a^{\frac{m}{n}}

  • anbn=a1nb1n=ab\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}} = \frac{a}{b}

Q: How do I simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, you can use the properties of radicals and exponents. You can rewrite each radical as a fractional exponent and then apply the quotient rule for exponents.

Q: Can you provide more examples of simplifying expressions with multiple radicals?

A: Yes, here are a few more examples:

  • 43x3⋅933x3=43x3⋅9333⋅x3=43x3⋅9333⋅1x3\frac{\sqrt[3]{4}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{9}}{\sqrt[3]{3x}} = \frac{\sqrt[3]{4}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{9}}{\sqrt[3]{3} \cdot \sqrt[3]{x}} = \frac{\sqrt[3]{4}}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{9}}{\sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{x}}

  • 1634x3⋅2535x3=16343⋅x3⋅25353⋅x3=16343⋅1x3⋅25353⋅1x3\frac{\sqrt[3]{16}}{\sqrt[3]{4x}} \cdot \frac{\sqrt[3]{25}}{\sqrt[3]{5x}} = \frac{\sqrt[3]{16}}{\sqrt[3]{4} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt[3]{25}}{\sqrt[3]{5} \cdot \sqrt[3]{x}} = \frac{\sqrt[3]{16}}{\sqrt[3]{4}} \cdot \frac{1}{\sqrt[3]{x}} \cdot \frac{\sqrt[3]{25}}{\sqrt[3]{5}} \cdot \frac{1}{\sqrt[3]{x}}