What Is The Value Of The Expression Below? 64 2 / 6 64^{2 / 6} 6 4 2/6 A. 4 B. 12 C. 2 3 \frac{2}{3} 3 2 ​ D. 64 3 \frac{64}{3} 3 64 ​

What is the Value of the Expression Below?

The value of an expression is a crucial aspect of mathematics, and it can be determined using various mathematical operations and rules. In this article, we will focus on evaluating the expression $64^{2 / 6}$ and determining its value.

Understanding Exponents and Fractions

Before we dive into evaluating the expression, it's essential to understand the concepts of exponents and fractions. Exponents are a shorthand way of representing repeated multiplication, and they can be used to simplify complex expressions. Fractions, on the other hand, are a way of representing part of a whole, and they can be used to simplify complex expressions as well.

Evaluating the Expression

To evaluate the expression $64^{2 / 6}$, we need to follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses: $2 / 6 = 1/3$
2. Raise 64 to the power of 1/3: $64^{1/3} = \sqrt[3]{64} = 4$
3. Raise 4 to the power of 1: $4^1 = 4$

Simplifying the Expression

Now that we have evaluated the expression, we can simplify it by multiplying the numerator and denominator by 3:

$64^{2 / 6} = 4^{1/3} = \frac{2}{3}$

Conclusion

In conclusion, the value of the expression $64^{2 / 6}$ is $\frac{2}{3}$. This is because we evaluated the expression using the order of operations (PEMDAS) and simplified it by multiplying the numerator and denominator by 3.

Why is this Important?

Understanding how to evaluate expressions like $64^{2 / 6}$ is crucial in mathematics, as it can help us solve complex problems and simplify complex expressions. It's also essential in real-world applications, such as finance, science, and engineering, where mathematical expressions are used to model and analyze complex systems.

Real-World Applications

The concept of evaluating expressions like $64^{2 / 6}$ has numerous real-world applications. For example:

• In finance, mathematical expressions are used to model and analyze complex financial systems, such as stock prices and interest rates.
• In science, mathematical expressions are used to model and analyze complex systems, such as population growth and chemical reactions.
• In engineering, mathematical expressions are used to design and optimize complex systems, such as bridges and buildings.

Tips and Tricks

Here are some tips and tricks to help you evaluate expressions like $64^{2 / 6}$:

• Always follow the order of operations (PEMDAS).
• Simplify complex expressions by multiplying the numerator and denominator by a common factor.
• Use exponent rules to simplify complex expressions.
• Practice, practice, practice! The more you practice evaluating expressions, the more comfortable you will become with the process.

Common Mistakes

Here are some common mistakes to avoid when evaluating expressions like $64^{2 / 6}$:

• Not following the order of operations (PEMDAS).
• Not simplifying complex expressions.
• Not using exponent rules to simplify complex expressions.
• Not practicing enough!

Conclusion

In conclusion, the value of the expression $64^{2 / 6}$ is $\frac{2}{3}$. This is because we evaluated the expression using the order of operations (PEMDAS) and simplified it by multiplying the numerator and denominator by 3. Understanding how to evaluate expressions like $64^{2 / 6}$ is crucial in mathematics, as it can help us solve complex problems and simplify complex expressions.
Frequently Asked Questions (FAQs) About Evaluating Expressions

In this article, we will answer some frequently asked questions about evaluating expressions like $64^{2 / 6}$.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when evaluating an expression. The acronym PEMDAS stands for:

• Parentheses: Evaluate expressions inside parentheses first.
• Exponents: Evaluate exponents next (e.g., $2^3$).
• Multiplication and Division: Evaluate multiplication and division operations from left to right.
• Addition and Subtraction: Evaluate addition and subtraction operations from left to right.

Q: How do I simplify complex expressions?

A: To simplify complex expressions, follow these steps:

1. Evaluate expressions inside parentheses.
2. Simplify exponents (e.g., $2^3$).
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of representing part of a whole, while a decimal is a way of representing a fraction as a numerical value. For example, the fraction $\frac{1}{2}$ is equal to the decimal 0.5.

Q: How do I evaluate expressions with negative exponents?

A: To evaluate expressions with negative exponents, follow these steps:

1. Rewrite the expression with a positive exponent.
2. Take the reciprocal of the expression.

For example, to evaluate the expression $2^{-3}$, we can rewrite it as $\frac{1}{2^3}$ and then simplify to $\frac{1}{8}$.

Q: What is the difference between an exponent and a power?

A: An exponent is a small number that is raised to a power, while a power is the result of raising a number to an exponent. For example, in the expression $2^3$, the 3 is an exponent and the 2 is the base.

Q: How do I evaluate expressions with multiple exponents?

A: To evaluate expressions with multiple exponents, follow these steps:

1. Evaluate the exponents from left to right.
2. Multiply the results together.

For example, to evaluate the expression $2^3 \cdot 3^2$, we can first evaluate the exponents to get $8 \cdot 9$ and then multiply the results together to get 72.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change, while a constant is a value that does not change. For example, in the expression $x + 5$, the $x$ is a variable and the 5 is a constant.

Q: How do I evaluate expressions with variables?

A: To evaluate expressions with variables, follow these steps:

1. Substitute the value of the variable into the expression.
2. Evaluate the expression using the order of operations (PEMDAS).

For example, to evaluate the expression $x + 5$ when $x = 3$, we can substitute the value of $x$ into the expression to get $3 + 5$ and then evaluate the expression to get 8.

Conclusion

In conclusion, evaluating expressions like $64^{2 / 6}$ requires a solid understanding of the order of operations (PEMDAS), exponent rules, and fraction simplification. By following these steps and practicing regularly, you can become proficient in evaluating complex expressions and solving mathematical problems with ease.