Simplifying the Fraction Further

Oct 13, 2025 by miodeya.com 20 views

Introduction

When dealing with fractions and radicals, it's essential to understand how to multiply them and simplify the result. In this article, we will explore the product of two fractions, one of which contains a radical, and express the result in its simplest radical form.

Understanding the Problem

The problem asks us to find the product of $\frac{6}{8}$ and $\frac{3\sqrt{}}{24}$ . To approach this problem, we need to understand the concept of multiplying fractions and radicals. When multiplying fractions, we simply multiply the numerators and denominators separately. However, when dealing with radicals, we need to apply the rules of radical multiplication.

Multiplying Fractions

To multiply fractions, we follow the rule:

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

In this case, we have:

\frac{6}{8} \cdot \frac{3\sqrt{}}{24} = \frac{6 \cdot 3\sqrt{}}{8 \cdot 24}

Simplifying the Fraction

Now, let's simplify the fraction by canceling out any common factors between the numerator and denominator.

\frac{6 \cdot 3\sqrt{}}{8 \cdot 24} = \frac{18\sqrt{}}{192}

Simplifying the Radical

To simplify the radical, we need to find the largest perfect square that divides the radicand (the number inside the radical). In this case, the radicand is $18$ . We can rewrite $18$ as $9 \cdot 2$ , where $9$ is a perfect square.

\frac{18\sqrt{}}{192} = \frac{9 \cdot 2 \cdot \sqrt{}}{192}

Simplifying the Fraction Further

Now, let's simplify the fraction further by canceling out any common factors between the numerator and denominator.

\frac{9 \cdot 2 \cdot \sqrt{}}{192} = \frac{9 \cdot \sqrt{}}{96}

Final Simplification

Finally, let's simplify the fraction by canceling out any common factors between the numerator and denominator.

\frac{9 \cdot \sqrt{}}{96} = \frac{\sqrt{}}{32/3}

Conclusion

In this article, we have explored the product of two fractions, one of which contains a radical, and expressed the result in its simplest radical form. We have applied the rules of fraction multiplication and radical simplification to arrive at the final answer.

Final Answer

The final answer is $\boxed{\frac{3\sqrt{}}{32}}$ .

Frequently Asked Questions

Q: What is the product of $\frac{6}{8}$ and $\frac{3\sqrt{}}{24}$ ? A: The product is $\frac{3\sqrt{}}{32}$ .
Q: How do I simplify a fraction with a radical? A: To simplify a fraction with a radical, you need to find the largest perfect square that divides the radicand and cancel out any common factors between the numerator and denominator.
Q: What is the rule for multiplying fractions? A: The rule for multiplying fractions is $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ .

References

[1] "Multiplying Fractions" by Math Open Reference
[2] "Radical Simplification" by Purplemath

Note: The references and related articles are fictional and for demonstration purposes only.

Introduction

Multiplying fractions and radicals can be a challenging task, especially when dealing with complex expressions. In this article, we will address some of the most frequently asked questions related to multiplying fractions and radicals.

Q: What is the rule for multiplying fractions?

A: The rule for multiplying fractions is:

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

This means that you multiply the numerators (a and c) together and the denominators (b and d) together.

Q: How do I simplify a fraction with a radical?

A: To simplify a fraction with a radical, you need to find the largest perfect square that divides the radicand and cancel out any common factors between the numerator and denominator.

For example, let's say you have the fraction $\frac{9\sqrt{}}{96}$ . To simplify this fraction, you can rewrite the radicand as $9 \cdot 2$ , where $9$ is a perfect square.

\frac{9\sqrt{}}{96} = \frac{9 \cdot 2 \cdot \sqrt{}}{96}

Then, you can cancel out the common factor of $9$ between the numerator and denominator.

\frac{9 \cdot 2 \cdot \sqrt{}}{96} = \frac{2 \cdot \sqrt{}}{32/3}

Q: What is the difference between multiplying fractions and multiplying radicals?

A: Multiplying fractions involves multiplying the numerators and denominators separately, while multiplying radicals involves applying the rules of radical multiplication.

For example, let's say you have the expression $\frac{2}{3} \cdot \sqrt{16}$ . To multiply this expression, you would first multiply the fraction $\frac{2}{3}$ by the radical $\sqrt{16}$ .

\frac{2}{3} \cdot \sqrt{16} = \frac{2 \cdot 4}{3}

Then, you would simplify the result.

\frac{2 \cdot 4}{3} = \frac{8}{3}

Q: Can I multiply a fraction by a radical?

A: Yes, you can multiply a fraction by a radical. To do this, you would follow the same rules as multiplying fractions and radicals separately.

For example, let's say you have the expression $\frac{2}{3} \cdot \sqrt{16}$ . To multiply this expression, you would first multiply the fraction $\frac{2}{3}$ by the radical $\sqrt{16}$ .

\frac{2}{3} \cdot \sqrt{16} = \frac{2 \cdot 4}{3}

Then, you would simplify the result.

\frac{2 \cdot 4}{3} = \frac{8}{3}

Q: How do I multiply two radicals?

A: To multiply two radicals, you would follow the rule:

\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}

For example, let's say you have the expression $\sqrt{16} \cdot \sqrt{9}$ . To multiply this expression, you would first multiply the radicands together.

\sqrt{16} \cdot \sqrt{9} = \sqrt{16 \cdot 9}

Then, you would simplify the result.

\sqrt{16 \cdot 9} = \sqrt{144} = 12

Q: Can I multiply a fraction by a radical with a different index?

A: Yes, you can multiply a fraction by a radical with a different index. To do this, you would follow the same rules as multiplying fractions and radicals separately.

For example, let's say you have the expression $\frac{2}{3} \cdot \sqrt[3]{16}$ . To multiply this expression, you would first multiply the fraction $\frac{2}{3}$ by the radical $\sqrt[3]{16}$ .

\frac{2}{3} \cdot \sqrt[3]{16} = \frac{2 \cdot 2}{3}

Then, you would simplify the result.

\frac{2 \cdot 2}{3} = \frac{4}{3}

Conclusion

Multiplying fractions and radicals can be a challenging task, but by following the rules and applying the concepts, you can simplify complex expressions and arrive at the correct solution.

Final Answer

The final answer is $\boxed{\frac{4}{3}}$ .

Frequently Asked Questions

Q: What is the rule for multiplying fractions? A: The rule for multiplying fractions is $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ .
Q: How do I simplify a fraction with a radical? A: To simplify a fraction with a radical, you need to find the largest perfect square that divides the radicand and cancel out any common factors between the numerator and denominator.
Q: What is the difference between multiplying fractions and multiplying radicals? A: Multiplying fractions involves multiplying the numerators and denominators separately, while multiplying radicals involves applying the rules of radical multiplication.

References

[1] "Multiplying Fractions" by Math Open Reference
[2] "Radical Simplification" by Purplemath