Simplify The Expression:$\[ \left(\frac{3}{x}\right)^4 \\]What Is The Simplified Expression In Terms Of $x$?

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Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this case, we're given the expression $\left(\frac{3}{x}\right)^4$ and asked to simplify it in terms of $x$. To approach this problem, we need to recall the rule for raising a fraction to a power.

Raising a Fraction to a Power

When a fraction is raised to a power, we raise both the numerator and the denominator to that power. In other words, if we have $\left(\frac{a}{b}\right)^n$, then we can rewrite it as $\frac{a^n}{b^n}$. This rule is crucial in simplifying the given expression.

Simplifying the Expression

Using the rule for raising a fraction to a power, we can rewrite the given expression as follows:

$$\left(\frac{3}{x}\right)^4 = \frac{3^4}{x^4}$$

Now, we can simplify the numerator and the denominator separately. The numerator is $3^4$, which is equal to $81$. The denominator is $x^4$, which is equal to $x$ multiplied by itself four times.

Simplifying the Numerator and Denominator

The numerator $3^4$ can be simplified as follows:

$$3^4 = 3 \times 3 \times 3 \times 3 = 81$$

The denominator $x^4$ can be simplified as follows:

$$x^4 = x \times x \times x \times x = x^4$$

Combining the Simplified Numerator and Denominator

Now that we have simplified the numerator and the denominator, we can combine them to get the final simplified expression:

$$\frac{3^4}{x^4} = \frac{81}{x^4}$$

Conclusion

In conclusion, the simplified expression $\left(\frac{3}{x}\right)^4$ is equal to $\frac{81}{x^4}$. This is the final answer in terms of $x$.

Final Answer

The final answer is $\boxed{\frac{81}{x^4}}$.

Understanding the Concept

To understand the concept of raising a fraction to a power, let's consider an example. Suppose we have the expression $\left(\frac{2}{3}\right)^2$. Using the rule for raising a fraction to a power, we can rewrite it as $\frac{2^2}{3^2}$. This simplifies to $\frac{4}{9}$.

Real-World Applications

The concept of raising a fraction to a power has numerous real-world applications. For instance, in finance, we often encounter expressions involving exponents, such as the compound interest formula. In physics, we use exponents to describe the behavior of particles and waves.

Common Mistakes

When simplifying expressions involving exponents, it's essential to remember the rule for raising a fraction to a power. A common mistake is to forget to raise both the numerator and the denominator to the power. For example, if we have $\left(\frac{2}{3}\right)^2$, we might mistakenly simplify it as $\frac{2}{3^2}$, which is incorrect.

Tips and Tricks

To simplify expressions involving exponents, follow these tips and tricks:

• Always raise both the numerator and the denominator to the power.
• Use the rule for raising a fraction to a power to simplify the expression.
• Simplify the numerator and the denominator separately.
• Combine the simplified numerator and denominator to get the final answer.

Practice Problems

To practice simplifying expressions involving exponents, try the following problems:

• Simplify the expression $\left(\frac{4}{5}\right)^3$.
• Simplify the expression $\left(\frac{2}{3}\right)^4$.
• Simplify the expression $\left(\frac{3}{4}\right)^2$.

Conclusion

In conclusion, simplifying expressions involving exponents requires a clear understanding of the rules governing exponents. By following the rule for raising a fraction to a power and simplifying the numerator and denominator separately, we can arrive at the final simplified expression. With practice and patience, you'll become proficient in simplifying expressions involving exponents.

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Frequently Asked Questions

Q: What is the rule for raising a fraction to a power?

A: When a fraction is raised to a power, we raise both the numerator and the denominator to that power. In other words, if we have $\left(\frac{a}{b}\right)^n$, then we can rewrite it as $\frac{a^n}{b^n}$.

Q: How do I simplify the expression $\left(\frac{3}{x}\right)^4$?

A: To simplify the expression $\left(\frac{3}{x}\right)^4$, we can use the rule for raising a fraction to a power. We raise both the numerator and the denominator to the power of 4, resulting in $\frac{3^4}{x^4}$. We can then simplify the numerator and the denominator separately to get the final answer.

Q: What is the final simplified expression for $\left(\frac{3}{x}\right)^4$?

A: The final simplified expression for $\left(\frac{3}{x}\right)^4$ is $\frac{81}{x^4}$.

Q: Can you provide an example of raising a fraction to a power?

A: Suppose we have the expression $\left(\frac{2}{3}\right)^2$. Using the rule for raising a fraction to a power, we can rewrite it as $\frac{2^2}{3^2}$. This simplifies to $\frac{4}{9}$.

Q: What are some real-world applications of raising a fraction to a power?

A: The concept of raising a fraction to a power has numerous real-world applications. For instance, in finance, we often encounter expressions involving exponents, such as the compound interest formula. In physics, we use exponents to describe the behavior of particles and waves.

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

A: When simplifying expressions involving exponents, it's essential to remember the rule for raising a fraction to a power. A common mistake is to forget to raise both the numerator and the denominator to the power. For example, if we have $\left(\frac{2}{3}\right)^2$, we might mistakenly simplify it as $\frac{2}{3^2}$, which is incorrect.

Q: Can you provide some tips and tricks for simplifying expressions involving exponents?

A: To simplify expressions involving exponents, follow these tips and tricks:

• Always raise both the numerator and the denominator to the power.
• Use the rule for raising a fraction to a power to simplify the expression.
• Simplify the numerator and the denominator separately.
• Combine the simplified numerator and denominator to get the final answer.

Q: Can you provide some practice problems for simplifying expressions involving exponents?

A: To practice simplifying expressions involving exponents, try the following problems:

• Simplify the expression $\left(\frac{4}{5}\right)^3$.
• Simplify the expression $\left(\frac{2}{3}\right)^4$.
• Simplify the expression $\left(\frac{3}{4}\right)^2$.

Q: What is the final answer to the expression $\left(\frac{3}{x}\right)^4$?

A: The final answer to the expression $\left(\frac{3}{x}\right)^4$ is $\boxed{\frac{81}{x^4}}$.

Conclusion

In conclusion, simplifying expressions involving exponents requires a clear understanding of the rules governing exponents. By following the rule for raising a fraction to a power and simplifying the numerator and denominator separately, we can arrive at the final simplified expression. With practice and patience, you'll become proficient in simplifying expressions involving exponents.