What Is The Value Of The Expression Below?$\left(64^3\right)^{1 / 6}$A. 32 B. 8 C. 4 D. 64

Understanding Exponents and Powers

In mathematics, exponents and powers are used to represent repeated multiplication of a number. The expression $\left(64^3\right)^{1 / 6}$ involves both exponents and powers, making it a bit complex to evaluate. To simplify this expression, we need to understand the rules of exponents and how to handle powers with fractional exponents.

The Rule of Exponents

The rule of exponents states that when we have a power raised to another power, we multiply the exponents. In other words, $(a^m)^n = a^{m \cdot n}$. This rule can be applied to simplify the given expression.

Simplifying the Expression

Let's start by simplifying the expression $\left(64^3\right)^{1 / 6}$. Using the rule of exponents, we can rewrite this expression as $64^{3 \cdot (1/6)}$. Now, we need to multiply the exponents, which gives us $64^{1/2}$.

Understanding Fractional Exponents

A fractional exponent is a way of expressing a power with a fraction as the exponent. In this case, $64^{1/2}$ can be read as "64 to the power of one-half". This means that we need to find the square root of 64.

Finding the Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we need to find the square root of 64. Since $8 \cdot 8 = 64$, the square root of 64 is 8.

Conclusion

In conclusion, the value of the expression $\left(64^3\right)^{1 / 6}$ is 8. This is because we simplified the expression using the rule of exponents and then found the square root of 64.

Why is this Important?

Understanding exponents and powers is crucial in mathematics, as it helps us simplify complex expressions and solve problems more efficiently. By applying the rule of exponents and understanding fractional exponents, we can evaluate expressions like $\left(64^3\right)^{1 / 6}$ and find their values.

Real-World Applications

Exponents and powers have many real-world applications, such as finance, science, and engineering. For example, in finance, exponents are used to calculate compound interest, while in science, exponents are used to describe the growth and decay of populations. In engineering, exponents are used to design and optimize systems.

Common Mistakes

When simplifying expressions with exponents and powers, it's easy to make mistakes. Some common mistakes include:

• Forgetting to multiply the exponents
• Not understanding the rules of exponents
• Not simplifying the expression correctly

Tips and Tricks

To simplify expressions with exponents and powers, follow these tips and tricks:

• Always multiply the exponents
• Understand the rules of exponents
• Simplify the expression step by step
• Check your work to ensure that the expression is simplified correctly

Conclusion

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Q: What is the rule of exponents?

A: The rule of exponents states that when we have a power raised to another power, we multiply the exponents. In other words, $(a^m)^n = a^{m \cdot n}$.

Q: How do I simplify an expression with exponents and powers?

A: To simplify an expression with exponents and powers, follow these steps:

1. Multiply the exponents
2. Simplify the expression step by step
3. Check your work to ensure that the expression is simplified correctly

Q: What is a fractional exponent?

A: A fractional exponent is a way of expressing a power with a fraction as the exponent. For example, $64^{1/2}$ can be read as "64 to the power of one-half".

Q: How do I find the square root of a number?

A: To find the square root of a number, you need to find a value that, when multiplied by itself, gives the original number. For example, the square root of 64 is 8 because $8 \cdot 8 = 64$.

Q: What are some common mistakes to avoid when simplifying expressions with exponents and powers?

A: Some common mistakes to avoid when simplifying expressions with exponents and powers include:

• Forgetting to multiply the exponents
• Not understanding the rules of exponents
• Not simplifying the expression correctly

Q: How do I apply the rule of exponents to simplify an expression?

A: To apply the rule of exponents, follow these steps:

1. Identify the power and the exponent
2. Multiply the exponents
3. Simplify the expression step by step

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

A: Exponents and powers have many real-world applications, such as:

• Finance: Exponents are used to calculate compound interest
• Science: Exponents are used to describe the growth and decay of populations
• Engineering: Exponents are used to design and optimize systems

Q: How do I check my work when simplifying an expression with exponents and powers?

A: To check your work when simplifying an expression with exponents and powers, follow these steps:

1. Simplify the expression step by step
2. Check your work to ensure that the expression is simplified correctly
3. Verify that the expression is equivalent to the original expression

Q: What are some tips and tricks for simplifying expressions with exponents and powers?

A: Some tips and tricks for simplifying expressions with exponents and powers include:

• Always multiply the exponents
• Understand the rules of exponents
• Simplify the expression step by step
• Check your work to ensure that the expression is simplified correctly

Conclusion

In conclusion, understanding exponents and powers is crucial in mathematics, as it helps us simplify complex expressions and solve problems more efficiently. By applying the rule of exponents and understanding fractional exponents, we can evaluate expressions like $\left(64^3\right)^{1 / 6}$ and find their values. Remember to always multiply the exponents, understand the rules of exponents, and simplify the expression step by step.