What Is 12 X 8 3 X 2 \frac{\sqrt{12 X^8}}{\sqrt{3 X^2}} 3 X 2 ​ 12 X 8 ​ ​ In Simplest Form, Where X ≥ 0 X \geq 0 X ≥ 0 ?A. 2 3 X 4 2 \sqrt{3} X^4 2 3 ​ X 4 B. 15 X 5 \sqrt{15} X^5 15 ​ X 5 C. 2 X 3 2 X^3 2 X 3 D. 2 X 2 2 X^2 2 X 2

Understanding the Problem

The given expression involves square roots and variables. To simplify the expression, we need to apply the rules of exponents and radicals. The expression is $\frac{\sqrt{12 x^8}}{\sqrt{3 x^2}}$, and we are asked to find its simplest form.

Simplifying the Expression

To simplify the expression, we can start by simplifying the square roots individually. We can rewrite $\sqrt{12}$ as $\sqrt{4 \cdot 3}$, which simplifies to $2\sqrt{3}$. Similarly, we can rewrite $\sqrt{3 x^2}$ as $\sqrt{3} \cdot \sqrt{x^2}$, which simplifies to $\sqrt{3} \cdot x$.

Applying the Quotient Rule for Radicals

Now that we have simplified the square roots, we can apply the quotient rule for radicals, which states that $\frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{a}}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{\sqrt{ab}}{b}$. In this case, we have $\frac{\sqrt{12 x^8}}{\sqrt{3 x^2}} = \frac{2\sqrt{3} \cdot x^4}{\sqrt{3} \cdot x}$.

Canceling Out Common Factors

We can now cancel out the common factors in the numerator and denominator. The $\sqrt{3}$ in the numerator and denominator cancel out, leaving us with $\frac{2x^4}{x}$. We can also cancel out one factor of $x$ in the numerator and denominator, leaving us with $2x^3$.

Conclusion

Therefore, the simplest form of the expression $\frac{\sqrt{12 x^8}}{\sqrt{3 x^2}}$ is $2x^3$.

Comparison with Answer Choices

Let's compare our answer with the answer choices:

• A. $2 \sqrt{3} x^4$ is not correct because we simplified the expression to $2x^3$, not $2 \sqrt{3} x^4$.
• B. $\sqrt{15} x^5$ is not correct because we simplified the expression to $2x^3$, not $\sqrt{15} x^5$.
• C. $2 x^3$ is correct because it matches our simplified expression.
• D. $2 x^2$ is not correct because we simplified the expression to $2x^3$, not $2 x^2$.

Final Answer

The final answer is C. $2 x^3$.

Understanding the Basics of Simplifying Expressions with Square Roots

In the previous article, we simplified the expression $\frac{\sqrt{12 x^8}}{\sqrt{3 x^2}}$ to its simplest form, which is $2x^3$. However, we may still have some questions about simplifying expressions with square roots. In this article, we will answer some frequently asked questions about simplifying expressions with square roots.

Q: What is the quotient rule for radicals?

A: The quotient rule for radicals states that $\frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{a}}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{\sqrt{ab}}{b}$. This rule allows us to simplify expressions with square roots by canceling out common factors.

Q: How do I simplify an expression with a square root in the numerator and a square root in the denominator?

A: To simplify an expression with a square root in the numerator and a square root in the denominator, we can apply the quotient rule for radicals. We can rewrite the expression as $\frac{\sqrt{a}}{\sqrt{b}} = \frac{\sqrt{a}}{\sqrt{b}} \cdot \frac{\sqrt{b}}{\sqrt{b}} = \frac{\sqrt{ab}}{b}$.

Q: What is the difference between simplifying an expression with a square root and simplifying an expression with a variable?

A: Simplifying an expression with a square root involves applying the rules of radicals, such as the quotient rule for radicals. Simplifying an expression with a variable involves applying the rules of exponents, such as the product rule for exponents.

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

A: To simplify an expression with multiple square roots, we can apply the rules of radicals, such as the product rule for radicals, which states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. We can also apply the quotient rule for radicals to simplify expressions with multiple square roots in the numerator and denominator.

Q: What is the final answer to the expression $\frac{\sqrt{12 x^8}}{\sqrt{3 x^2}}$?

A: The final answer to the expression $\frac{\sqrt{12 x^8}}{\sqrt{3 x^2}}$ is $2x^3$.

Q: Why is it important to simplify expressions with square roots?

A: Simplifying expressions with square roots is important because it allows us to write expressions in their simplest form, which can make it easier to solve equations and inequalities. Simplifying expressions with square roots also helps us to avoid errors and make calculations more efficient.

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

A: Yes, here are a few more examples of simplifying expressions with square roots:

• $\frac{\sqrt{20 x^6}}{\sqrt{5 x^2}} = \frac{2\sqrt{2} \cdot x^3}{\sqrt{5} \cdot x} = \frac{2\sqrt{2} \cdot x^2}{\sqrt{5}}$
• $\frac{\sqrt{15 x^8}}{\sqrt{3 x^4}} = \frac{3\sqrt{5} \cdot x^4}{\sqrt{3} \cdot x^2} = 3\sqrt{5} \cdot x^2$
• $\frac{\sqrt{24 x^10}}{\sqrt{6 x^6}} = \frac{2\sqrt{6} \cdot x^5}{\sqrt{6} \cdot x^3} = 2x^2$

Conclusion

Simplifying expressions with square roots is an important skill in algebra and mathematics. By applying the rules of radicals, such as the quotient rule for radicals, we can simplify expressions with square roots and write them in their simplest form. We hope that this article has helped to clarify any questions you may have had about simplifying expressions with square roots.