What Is The Following Product? Assume X ≥ 0 X \geq 0 X ≥ 0 . X 2 3 ⋅ X 3 4 \sqrt[3]{x^2} \cdot \sqrt[4]{x^3} 3 X 2 ​ ⋅ 4 X 3 ​ A. X X X \sqrt{x} X X ​ B. X 5 12 \sqrt[12]{x^5} 12 X 5 ​ C. X\left(\sqrt[12]{x^5}\right ] D. X 6 X^6 X 6

Understanding the Problem and Solution

When dealing with mathematical expressions involving exponents and roots, it's essential to understand the properties of these operations to simplify and solve the problem effectively. In this case, we're given the expression $\sqrt[3]{x^2} \cdot \sqrt[4]{x^3}$ and asked to simplify it.

Breaking Down the Expression

To simplify the given expression, we need to apply the properties of exponents and roots. The expression involves two terms: $\sqrt[3]{x^2}$ and $\sqrt[4]{x^3}$. We can start by simplifying each term separately.

Simplifying the First Term

The first term is $\sqrt[3]{x^2}$. To simplify this, we can use the property of exponents that states $\sqrt[n]{x^n} = x^{\frac{n}{n}} = x$. In this case, we have $\sqrt[3]{x^2} = x^{\frac{2}{3}}$.

Simplifying the Second Term

The second term is $\sqrt[4]{x^3}$. To simplify this, we can use the property of exponents that states $\sqrt[n]{x^n} = x^{\frac{n}{n}} = x$. In this case, we have $\sqrt[4]{x^3} = x^{\frac{3}{4}}$.

Combining the Terms

Now that we have simplified each term, we can combine them to get the final expression. We multiply the two terms together, which gives us:

$x^{\frac{2}{3}} \cdot x^{\frac{3}{4}}$

Applying the Product of Powers Property

To simplify the expression further, we can use the product of powers property, which states that $x^a \cdot x^b = x^{a+b}$. Applying this property to our expression, we get:

$x^{\frac{2}{3} + \frac{3}{4}}$

Simplifying the Exponent

To simplify the exponent, we need to find a common denominator. The least common multiple of 3 and 4 is 12, so we can rewrite the exponent as:

$x^{\frac{8}{12} + \frac{9}{12}}$

Combining the Fractions

Now that we have a common denominator, we can combine the fractions in the exponent:

$x^{\frac{17}{12}}$

Conclusion

In conclusion, the simplified expression is $\boxed{x^{\frac{17}{12}}}$. This is the final answer to the problem.

Comparison with the Options

Let's compare our answer with the options provided:

A. $x \sqrt{x}$

This option is not equivalent to our answer.

B. $\sqrt[12]{x^5}$

This option is not equivalent to our answer.

C. $x\left(\sqrt[12]{x^5}\right)$

This option is not equivalent to our answer.

D. $x^6$

This option is not equivalent to our answer.

Final Answer

The final answer is $\boxed{x^{\frac{17}{12}}}$.

Understanding the Problem and Solution

In the previous article, we simplified the expression $\sqrt[3]{x^2} \cdot \sqrt[4]{x^3}$ to get the final answer of $x^{\frac{17}{12}}$. However, we received several questions from readers who were unsure about the steps involved in simplifying the expression. In this article, we will address some of the most frequently asked questions (FAQs) about the product of roots.

Q: What is the property of exponents that we used to simplify the first term?

A: The property of exponents that we used to simplify the first term is $\sqrt[n]{x^n} = x^{\frac{n}{n}} = x$. This property states that the nth root of a number raised to the power of n is equal to the number itself.

Q: How do we simplify the second term?

A: To simplify the second term, we use the property of exponents that states $\sqrt[n]{x^n} = x^{\frac{n}{n}} = x$. In this case, we have $\sqrt[4]{x^3} = x^{\frac{3}{4}}$.

Q: What is the product of powers property?

A: The product of powers property states that $x^a \cdot x^b = x^{a+b}$. This property allows us to combine two terms with the same base by adding their exponents.

Q: How do we simplify the exponent?

A: To simplify the exponent, we need to find a common denominator. The least common multiple of 3 and 4 is 12, so we can rewrite the exponent as $x^{\frac{8}{12} + \frac{9}{12}}$. We can then combine the fractions in the exponent to get $x^{\frac{17}{12}}$.

Q: What is the final answer?

A: The final answer is $\boxed{x^{\frac{17}{12}}}$. This is the simplified expression of $\sqrt[3]{x^2} \cdot \sqrt[4]{x^3}$.

Q: How does the final answer compare to the options provided?

A: The final answer does not match any of the options provided. The options are A. $x \sqrt{x}$, B. $\sqrt[12]{x^5}$, C. $x\left(\sqrt[12]{x^5}\right)$, and D. $x^6$. None of these options are equivalent to the final answer of $x^{\frac{17}{12}}$.

Q: What is the significance of the exponent $\frac{17}{12}$?

A: The exponent $\frac{17}{12}$ represents the power to which the variable $x$ is raised. In this case, the variable $x$ is raised to the power of $\frac{17}{12}$.

Q: How can I apply this concept to other problems?

A: You can apply this concept to other problems by using the properties of exponents and roots. Remember to simplify the expression by combining like terms and using the product of powers property.

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

A: Some common mistakes to avoid when simplifying expressions include:

• Not using the product of powers property
• Not finding a common denominator when simplifying the exponent
• Not combining like terms
• Not checking the final answer against the options provided

Conclusion

In conclusion, the product of roots is a fundamental concept in mathematics that involves simplifying expressions with exponents and roots. By understanding the properties of exponents and roots, you can simplify complex expressions and arrive at the final answer. Remember to use the product of powers property, find a common denominator, and combine like terms to simplify the expression.