Simplify The Expression: $\[ \frac{5 \sqrt[4]{2}}{4 \sqrt[4]{162}} \\]

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. One common type of expression that requires simplification is the one with a radical in the denominator. In this article, we will focus on simplifying the expression $\frac{5 \sqrt[4]{2}}{4 \sqrt[4]{162}}$ by rationalizing the denominator.

Understanding the Problem

The given expression is a fraction with a fourth root in the numerator and the denominator. To simplify this expression, we need to rationalize the denominator, which means removing the radical from the denominator. This can be achieved by multiplying both the numerator and the denominator by a suitable expression that will eliminate the radical in the denominator.

Rationalizing the Denominator

To rationalize the denominator, we need to find an expression that, when multiplied with the denominator, will eliminate the fourth root. Since the denominator is $\sqrt[4]{162}$, we can try to find an expression that, when multiplied with the denominator, will result in a perfect fourth power.

Step 1: Factorize the Denominator

The first step is to factorize the denominator, $\sqrt[4]{162}$. We can start by finding the prime factorization of 162.

$$162 = 2 \times 81 = 2 \times 9^2 = 2 \times (3^2)^2 = 2 \times 3^4$$

Now, we can rewrite the denominator as:

$$\sqrt[4]{162} = \sqrt[4]{2 \times 3^4} = \sqrt[4]{2} \times \sqrt[4]{3^4} = \sqrt[4]{2} \times 3$$

Step 2: Multiply by the Conjugate

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt[4]{2} \times 3$ is $\sqrt[4]{2} \times 3$ itself.

$$\frac{5 \sqrt[4]{2}}{4 \sqrt[4]{162}} = \frac{5 \sqrt[4]{2}}{4 \sqrt[4]{2} \times 3}$$

Now, we can multiply both the numerator and the denominator by $\sqrt[4]{2} \times 3$.

$$\frac{5 \sqrt[4]{2}}{4 \sqrt[4]{2} \times 3} = \frac{5 \sqrt[4]{2} \times \sqrt[4]{2} \times 3}{4 \sqrt[4]{2} \times \sqrt[4]{2} \times 3}$$

Step 3: Simplify the Expression

Now, we can simplify the expression by canceling out the common factors in the numerator and the denominator.

$$\frac{5 \sqrt[4]{2} \times \sqrt[4]{2} \times 3}{4 \sqrt[4]{2} \times \sqrt[4]{2} \times 3} = \frac{5 \times 2 \times 3}{4 \times 2 \times 3}$$

Final Answer

The final answer is:

$$\frac{5 \times 2 \times 3}{4 \times 2 \times 3} = \frac{30}{24} = \frac{5}{4}$$

Conclusion

In this article, we simplified the expression $\frac{5 \sqrt[4]{2}}{4 \sqrt[4]{162}}$ by rationalizing the denominator. We factorized the denominator, multiplied by the conjugate, and simplified the expression to get the final answer. This problem requires a good understanding of radicals and rationalizing the denominator, which is an essential skill in mathematics.

Common Mistakes to Avoid

When simplifying expressions with radicals, it's essential to avoid common mistakes such as:

• Not rationalizing the denominator
• Not simplifying the expression properly
• Not canceling out common factors

By following the steps outlined in this article, you can avoid these common mistakes and simplify expressions with radicals efficiently.

Real-World Applications

Simplifying expressions with radicals has many real-world applications, such as:

• Physics: Simplifying expressions with radicals is essential in physics, particularly in problems involving waves and vibrations.
• Engineering: Simplifying expressions with radicals is crucial in engineering, particularly in problems involving electrical circuits and mechanical systems.
• Computer Science: Simplifying expressions with radicals is essential in computer science, particularly in problems involving algorithms and data structures.

Introduction

In our previous article, we simplified the expression $\frac{5 \sqrt[4]{2}}{4 \sqrt[4]{162}}$ by rationalizing the denominator. In this article, we will answer some frequently asked questions about rationalizing the denominator and provide additional examples to help you master this skill.

Q&A

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is the process of removing the radical from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by a suitable expression that will eliminate the radical in the denominator.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify the expression and make it easier to work with. Rationalizing the denominator also helps to eliminate any radicals in the denominator, which can make the expression more manageable.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, you need to follow these steps:

1. Factorize the denominator
2. Multiply by the conjugate
3. Simplify the expression

Q: What is the conjugate?

A: The conjugate of a binomial expression is the expression with the opposite sign between the two terms. For example, the conjugate of $a + b$ is $a - b$.

Q: How do I find the conjugate?

A: To find the conjugate, you need to change the sign between the two terms. For example, the conjugate of $\sqrt[4]{2} \times 3$ is $\sqrt[4]{2} \times 3$ itself.

Q: Can I rationalize the denominator of a fraction with a square root?

A: Yes, you can rationalize the denominator of a fraction with a square root. The process is similar to rationalizing the denominator of a fraction with a fourth root.

Q: Can I rationalize the denominator of a fraction with a cube root?

A: Yes, you can rationalize the denominator of a fraction with a cube root. The process is similar to rationalizing the denominator of a fraction with a fourth root.

Q: How do I simplify the expression after rationalizing the denominator?

A: To simplify the expression after rationalizing the denominator, you need to cancel out any common factors in the numerator and the denominator.

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

• Not rationalizing the denominator
• Not simplifying the expression properly
• Not canceling out common factors

Q: How do I apply rationalizing the denominator in real-world problems?

A: Rationalizing the denominator is essential in various real-world problems, such as physics, engineering, and computer science. By mastering the skill of rationalizing the denominator, you can apply it to various real-world problems and become a proficient mathematician.

Additional Examples

Example 1: Rationalizing the Denominator of a Fraction with a Square Root

Simplify the expression $\frac{3 \sqrt{2}}{4 \sqrt{8}}$ by rationalizing the denominator.

Solution:

$$\frac{3 \sqrt{2}}{4 \sqrt{8}} = \frac{3 \sqrt{2}}{4 \sqrt{4 \times 2}} = \frac{3 \sqrt{2}}{4 \times 2 \sqrt{2}} = \frac{3}{4 \times 2} = \frac{3}{8}$$

Example 2: Rationalizing the Denominator of a Fraction with a Cube Root

Simplify the expression $\frac{2 \sqrt[3]{3}}{3 \sqrt[3]{27}}$ by rationalizing the denominator.

Solution:

$$\frac{2 \sqrt[3]{3}}{3 \sqrt[3]{27}} = \frac{2 \sqrt[3]{3}}{3 \sqrt[3]{3^3}} = \frac{2 \sqrt[3]{3}}{3 \times 3 \sqrt[3]{3}} = \frac{2}{3 \times 3} = \frac{2}{9}$$

Conclusion

In this article, we answered some frequently asked questions about rationalizing the denominator and provided additional examples to help you master this skill. By following the steps outlined in this article, you can simplify expressions with radicals efficiently and become a proficient mathematician.