Evaluate The Expression:${ \left(\frac{2}{y}\right)^4 }$

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Introduction

In mathematics, expressions involving exponents and fractions can be challenging to evaluate. The given expression, $\left(\frac{2}{y}\right)^4$, requires us to apply the rules of exponents and fraction manipulation. In this article, we will break down the expression and provide a step-by-step solution to evaluate it.

Understanding Exponents and Fractions

Before we dive into the solution, let's review the rules of exponents and fractions. When a fraction is raised to a power, we can apply the exponent to both the numerator and the denominator. This is known as the power rule for fractions.

Power Rule for Fractions:

If $a$ and $b$ are real numbers and $n$ is an integer, then:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Evaluating the Expression

Now that we have reviewed the power rule for fractions, let's apply it to the given expression:

$\left(\frac{2}{y}\right)^4$

Using the power rule, we can rewrite the expression as:

$\frac{2^4}{y^4}$

Simplifying the Expression

The next step is to simplify the expression by evaluating the exponent on the numerator and the denominator.

Simplifying the Numerator:

$2^4 = 16$

Simplifying the Denominator:

$y^4$ remains the same, as it is already in its simplest form.

Final Expression

Now that we have simplified the numerator and the denominator, we can write the final expression:

$\frac{16}{y^4}$

Conclusion

In this article, we evaluated the expression $\left(\frac{2}{y}\right)^4$ using the power rule for fractions. We simplified the expression by evaluating the exponent on the numerator and the denominator, resulting in the final expression $\frac{16}{y^4}$. This demonstrates the importance of understanding and applying the rules of exponents and fractions in mathematical expressions.

Additional Examples

To further illustrate the concept, let's consider a few additional examples:

Example 1:

Evaluate the expression $\left(\frac{3}{x}\right)^3$

Using the power rule, we can rewrite the expression as:

$\frac{3^3}{x^3}$

Simplifying the numerator and the denominator, we get:

$\frac{27}{x^3}$

Example 2:

Evaluate the expression $\left(\frac{4}{z}\right)^2$

Using the power rule, we can rewrite the expression as:

$\frac{4^2}{z^2}$

Simplifying the numerator and the denominator, we get:

$\frac{16}{z^2}$

Common Mistakes to Avoid

When evaluating expressions involving exponents and fractions, it's essential to avoid common mistakes. Here are a few examples:

• Not applying the power rule: Failing to apply the power rule for fractions can lead to incorrect results.
• Not simplifying the numerator and denominator: Failing to simplify the numerator and denominator can result in a more complex expression than necessary.
• Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect results.

Real-World Applications

Expressions involving exponents and fractions have numerous real-world applications. Here are a few examples:

• Physics: Exponents and fractions are used to describe the motion of objects, including velocity, acceleration, and force.
• Engineering: Exponents and fractions are used to describe the behavior of electrical circuits, including resistance, capacitance, and inductance.
• Economics: Exponents and fractions are used to describe the behavior of economic systems, including interest rates, inflation, and economic growth.

Final Thoughts

In conclusion, evaluating expressions involving exponents and fractions requires a deep understanding of the rules of exponents and fractions. By applying the power rule for fractions and simplifying the numerator and denominator, we can arrive at the final expression. This article has demonstrated the importance of understanding and applying the rules of exponents and fractions in mathematical expressions.

Introduction

In our previous article, we evaluated the expression $\left(\frac{2}{y}\right)^4$ using the power rule for fractions. In this article, we will address some common questions and concerns related to evaluating expressions involving exponents and fractions.

Q&A

Q1: What is the power rule for fractions?

A1: The power rule for fractions states that if $a$ and $b$ are real numbers and $n$ is an integer, then:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Q2: How do I apply the power rule for fractions?

A2: To apply the power rule for fractions, simply raise the numerator and denominator to the power indicated. For example, if we have the expression $\left(\frac{2}{y}\right)^4$, we can rewrite it as:

$\frac{2^4}{y^4}$

Q3: What is the difference between a fraction and an exponent?

A3: A fraction is a way of expressing a part of a whole, while an exponent is a way of expressing repeated multiplication. For example, the fraction $\frac{1}{2}$ represents one half of a whole, while the exponent $2^3$ represents $2$ multiplied by itself $3$ times.

Q4: How do I simplify an expression involving exponents and fractions?

A4: To simplify an expression involving exponents and fractions, follow these steps:

1. Apply the power rule for fractions.
2. Simplify the numerator and denominator separately.
3. Combine the simplified numerator and denominator.

Q5: What are some common mistakes to avoid when evaluating expressions involving exponents and fractions?

A5: Some common mistakes to avoid when evaluating expressions involving exponents and fractions include:

• Not applying the power rule for fractions.
• Not simplifying the numerator and denominator.
• Not following the order of operations (PEMDAS).

Q6: How do I evaluate an expression with a negative exponent?

A6: To evaluate an expression with a negative exponent, follow these steps:

1. Rewrite the expression with a positive exponent by taking the reciprocal of the fraction.
2. Apply the power rule for fractions.
3. Simplify the numerator and denominator.

Q7: Can I use the power rule for fractions with decimal numbers?

A7: Yes, you can use the power rule for fractions with decimal numbers. For example, if we have the expression $\left(\frac{0.5}{y}\right)^4$, we can rewrite it as:

$\frac{(0.5)^4}{y^4}$

Q8: How do I evaluate an expression with a variable in the exponent?

A8: To evaluate an expression with a variable in the exponent, follow these steps:

1. Apply the power rule for fractions.
2. Simplify the numerator and denominator separately.
3. Combine the simplified numerator and denominator.

Q9: Can I use the power rule for fractions with complex numbers?

A9: Yes, you can use the power rule for fractions with complex numbers. For example, if we have the expression $\left(\frac{2+3i}{y}\right)^4$, we can rewrite it as:

$\frac{(2+3i)^4}{y^4}$

Q10: How do I evaluate an expression with multiple exponents?

A10: To evaluate an expression with multiple exponents, follow these steps:

1. Apply the power rule for fractions to each exponent separately.
2. Simplify the numerator and denominator separately.
3. Combine the simplified numerator and denominator.

Conclusion

In this article, we have addressed some common questions and concerns related to evaluating expressions involving exponents and fractions. By following the power rule for fractions and simplifying the numerator and denominator, we can arrive at the final expression. We hope this article has been helpful in clarifying any doubts you may have had.