Consider The Expression:$\frac 6 7}{6 4}$1. Express Each Exponent In Expanded Form $\frac{6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 {6 \cdot 6 \cdot 6 \cdot 6}$2. Divide Common Factors:What Is The Exponent Of The Simplified

Introduction

In mathematics, exponential expressions are a fundamental concept that helps us represent large numbers in a more manageable form. When dealing with exponential expressions, it's essential to understand the rules of simplification to avoid errors and arrive at the correct solution. In this article, we'll explore the process of simplifying exponential expressions using the given example: $\frac{6^7}{6^4}$.

Step 1: Express Each Exponent in Expanded Form

To simplify the given expression, we need to express each exponent in expanded form. This involves multiplying the base number (6) by itself as many times as the exponent indicates.

Expanded Form of $6^7$

The expanded form of $6^7$ is:

$$6^7 = 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6$$

Expanded Form of $6^4$

The expanded form of $6^4$ is:

$$6^4 = 6 \cdot 6 \cdot 6 \cdot 6$$

Step 2: Divide Common Factors

Now that we have the expanded forms of both exponents, we can simplify the expression by dividing common factors. When dividing exponential expressions with the same base, we subtract the exponents.

Simplifying the Expression

Using the expanded forms from Step 1, we can rewrite the original expression as:

$$\frac{6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6}{6 \cdot 6 \cdot 6 \cdot 6}$$

To simplify this expression, we can divide the numerator and denominator by the common factors:

$$\frac{6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6}{6 \cdot 6 \cdot 6 \cdot 6} = \frac{6^7}{6^4}$$

Subtracting Exponents

Since the base numbers are the same, we can subtract the exponents:

$$\frac{6^7}{6^4} = 6^{7-4} = 6^3$$

Conclusion

In this article, we've explored the process of simplifying exponential expressions using the given example: $\frac{6^7}{6^4}$. By expressing each exponent in expanded form and dividing common factors, we arrived at the simplified expression: $6^3$. This demonstrates the importance of understanding the rules of simplification when working with exponential expressions.

Real-World Applications

Simplifying exponential expressions has numerous real-world applications in various fields, including:

• Science: Exponential expressions are used to model population growth, chemical reactions, and other scientific phenomena.
• Finance: Exponential expressions are used to calculate compound interest, investment returns, and other financial metrics.
• Engineering: Exponential expressions are used to model complex systems, optimize performance, and predict outcomes.

Common Mistakes to Avoid

When simplifying exponential expressions, it's essential to avoid common mistakes, including:

• Incorrectly subtracting exponents: Make sure to subtract the exponents only when the base numbers are the same.
• Failing to simplify: Don't forget to simplify the expression by dividing common factors and subtracting exponents.

Final Thoughts

Introduction

In our previous article, we explored the process of simplifying exponential expressions using the example: $\frac{6^7}{6^4}$. We expressed each exponent in expanded form, divided common factors, and subtracted exponents to arrive at the simplified expression: $6^3$. In this article, we'll answer some frequently asked questions about simplifying exponential expressions.

Q&A

Q: What is the rule for simplifying exponential expressions?

A: The rule for simplifying exponential expressions is to express each exponent in expanded form, divide common factors, and subtract exponents when the base numbers are the same.

Q: How do I express an exponent in expanded form?

A: To express an exponent in expanded form, multiply the base number by itself as many times as the exponent indicates. For example, $6^7$ can be expressed in expanded form as $6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6$.

Q: What is the difference between $6^7$ and $6^4$?

A: $6^7$ is equal to $6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6$, while $6^4$ is equal to $6 \cdot 6 \cdot 6 \cdot 6$. The difference is that $6^7$ has one more factor of 6 than $6^4$.

Q: How do I divide common factors in exponential expressions?

A: To divide common factors in exponential expressions, divide the numerator and denominator by the common factors. For example, $\frac{6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6}{6 \cdot 6 \cdot 6 \cdot 6}$ can be simplified by dividing the numerator and denominator by $6 \cdot 6 \cdot 6 \cdot 6$.

Q: What is the result of subtracting exponents in exponential expressions?

A: When subtracting exponents in exponential expressions, the result is the difference between the two exponents. For example, $6^{7-4}$ is equal to $6^3$.

Q: Can I simplify exponential expressions with different base numbers?

A: No, you cannot simplify exponential expressions with different base numbers. The rule for simplifying exponential expressions only applies when the base numbers are the same.

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

A: Some common mistakes to avoid when simplifying exponential expressions include:

• Incorrectly subtracting exponents
• Failing to simplify the expression
• Not expressing each exponent in expanded form
• Not dividing common factors

Real-World Applications

Simplifying exponential expressions has numerous real-world applications in various fields, including:

• Science: Exponential expressions are used to model population growth, chemical reactions, and other scientific phenomena.
• Finance: Exponential expressions are used to calculate compound interest, investment returns, and other financial metrics.
• Engineering: Exponential expressions are used to model complex systems, optimize performance, and predict outcomes.

Conclusion

In this article, we've answered some frequently asked questions about simplifying exponential expressions. By understanding the rules of simplification and avoiding common mistakes, you'll be able to tackle complex problems with confidence. Remember to express each exponent in expanded form, divide common factors, and subtract exponents to arrive at the simplified expression.

Final Thoughts

Simplifying exponential expressions is a crucial skill in mathematics that has numerous real-world applications. By mastering this skill, you'll be able to solve complex problems and make informed decisions in various fields.