Which Symbol Correctly Completes The Statement?$\[ 4^8 \, ? \, 8^4 \\]A. \[$\ \textless \ \$\]B. \[$\ \textgreater \ \$\]C. \[$=\$\]D. None Of These

Introduction

In mathematics, exponents and inequalities are fundamental concepts that help us understand and solve various problems. Exponents represent repeated multiplication of a number, while inequalities compare values to determine their relationships. In this article, we will explore the concept of exponents and inequalities, focusing on the given statement: ${ 4^8 , ? , 8^4 }$. We will analyze the options and determine which symbol correctly completes the statement.

Exponents

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $4^8$ represents $4$ multiplied by itself $8$ times: $4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$. This can be calculated as $4^8 = 65536$.

Inequalities

Inequalities are used to compare values and determine their relationships. There are four basic inequality symbols:

• Less than ($<$)
• Greater than ($>$)
• Less than or equal to ($\leq$)
• Greater than or equal to ($\geq$)

Analyzing the Statement

The given statement is ${ 4^8 , ? , 8^4 }$. To determine which symbol correctly completes the statement, we need to calculate the values of $4^8$ and $8^4$.

Calculating Exponents

Let's calculate the values of $4^8$ and $8^4$:

• $4^8 = 65536$
• $8^4 = 4096$

Comparing Values

Now that we have calculated the values of $4^8$ and $8^4$, we can compare them to determine the correct symbol.

• $4^8 = 65536$ is greater than $8^4 = 4096$

Determining the Correct Symbol

Based on the comparison, we can determine that the correct symbol is the greater than symbol ($>$).

Conclusion

In conclusion, the correct symbol that completes the statement ${ 4^8 , ? , 8^4 }$ is the greater than symbol ($>$). This is because $4^8 = 65536$ is greater than $8^4 = 4096$.

Final Answer

The final answer is:

• B. {\ \textgreater \ $}$

Additional Examples

Here are some additional examples to illustrate the concept:

• $2^5 = 32$ and $5^2 = 25$, so $2^5 > 5^2$
• $3^4 = 81$ and $4^3 = 64$, so $3^4 > 4^3$
• $5^3 = 125$ and $3^5 = 243$, so $5^3 < 3^5$

These examples demonstrate how to compare values using exponents and inequalities.

Common Mistakes

Here are some common mistakes to avoid when working with exponents and inequalities:

• Confusing the order of operations: Make sure to follow the order of operations (PEMDAS) when working with exponents and inequalities.
• Not calculating exponents correctly: Double-check your calculations to ensure that you are getting the correct values.
• Not comparing values correctly: Make sure to compare values using the correct inequality symbols.

By following these tips and examples, you can improve your understanding of exponents and inequalities and become more confident in your math skills.

Conclusion

$$$$$$

Introduction

In our previous article, we explored the concept of exponents and inequalities, focusing on the statement ${ 4^8 , ? , 8^4 }$. We determined that the correct symbol that completes the statement is the greater than symbol ($>$). In this article, we will provide a Q&A guide to help you better understand exponents and inequalities.

Q: What is an exponent?

A: An exponent is a shorthand way of representing repeated multiplication of a number. For example, $4^8$ represents $4$ multiplied by itself $8$ times: $4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4$.

Q: How do I calculate exponents?

A: To calculate exponents, you can use the following formula:

$a^b = a \times a \times a \times ... \times a$ (b times)

For example, $4^8 = 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 65536$.

Q: What is an inequality?

A: An inequality is a statement that compares two values to determine their relationship. There are four basic inequality symbols:

• Less than ($<$)
• Greater than ($>$)
• Less than or equal to ($\leq$)
• Greater than or equal to ($\geq$)

Q: How do I compare values using inequalities?

A: To compare values using inequalities, you need to determine the relationship between the two values. For example:

• If $a > b$, then $a$ is greater than $b$.
• If $a < b$, then $a$ is less than $b$.
• If $a \geq b$, then $a$ is greater than or equal to $b$.
• If $a \leq b$, then $a$ is less than or equal to $b$.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when working with mathematical expressions. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I avoid common mistakes when working with exponents and inequalities?

A: To avoid common mistakes when working with exponents and inequalities, make sure to:

• Follow the order of operations (PEMDAS).
• Calculate exponents correctly.
• Compare values correctly using the correct inequality symbols.

Q: What are some real-world applications of exponents and inequalities?

A: Exponents and inequalities have many real-world applications, including:

• Finance: Exponents are used to calculate interest rates and investments.
• Science: Inequalities are used to compare values and determine relationships in scientific experiments.
• Engineering: Exponents and inequalities are used to design and optimize systems.

Conclusion

In conclusion, exponents and inequalities are fundamental concepts in mathematics that have many real-world applications. By understanding exponents and inequalities, you can solve a wide range of math problems and become more confident in your math skills. We hope this Q&A guide has helped you better understand exponents and inequalities.

Additional Resources

For more information on exponents and inequalities, check out the following resources:

• Khan Academy: Exponents and Inequalities
• Mathway: Exponents and Inequalities
• Wolfram Alpha: Exponents and Inequalities

Final Answer

The final answer is:

• B. {\ \textgreater \ $}$

Common Mistakes

Here are some common mistakes to avoid when working with exponents and inequalities:

• Confusing the order of operations: Make sure to follow the order of operations (PEMDAS) when working with exponents and inequalities.
• Not calculating exponents correctly: Double-check your calculations to ensure that you are getting the correct values.
• Not comparing values correctly: Make sure to compare values using the correct inequality symbols.

By following these tips and examples, you can improve your understanding of exponents and inequalities and become more confident in your math skills.