Which Expression Is Equivalent To 64 A 6 \sqrt{64 A^6} 64 A 6 ​ ?A. 8 ∣ A 3 ∣ 8\left|a^3\right| 8 ​ A 3 ​ B. 8 A 6 8 A^6 8 A 6 C. 8 ∣ A 3 ∣ \sqrt{8\left|a^3\right|} 8 ∣ A 3 ∣ ​ D. 8 A 6 \sqrt{8 A^6} 8 A 6 ​

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\sqrt{64 a^6}$. We will examine each option and determine which one is equivalent to the given expression.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a root or a radical sign. The most common radical sign is the square root sign, denoted by $\sqrt{}$. Radical expressions can be simplified by factoring out perfect squares or using the properties of radicals.

Simplifying the Given Expression

To simplify the given expression $\sqrt{64 a^6}$, we need to factor out perfect squares. We can start by breaking down the number 64 into its prime factors.

64 = 2^6

Now, we can rewrite the expression as:

$\sqrt{2^6 a^6}$

Next, we can use the property of radicals that states $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$. Applying this property, we get:

$\sqrt{2^6 a^6} = \sqrt{2^6} \cdot \sqrt{a^6}$

Now, we can simplify each radical expression separately.

$\sqrt{2^6} = 2^3 = 8$

$\sqrt{a^6} = a^3$

Therefore, the simplified expression is:

$8 a^3$

However, we need to consider the absolute value of $a^3$, as the given options include an absolute value sign.

Evaluating the Options

Now, let's evaluate each option and determine which one is equivalent to the simplified expression.

Option A: $8\left|a^3\right|$

This option includes an absolute value sign, which is consistent with our simplified expression. However, we need to consider whether the absolute value sign is necessary.

Option B: $8 a^6$

This option is not equivalent to the simplified expression, as it does not include the absolute value sign.

Option C: $\sqrt{8\left|a^3\right|}$

This option includes a square root sign, which is not necessary in this case.

Option D: $\sqrt{8 a^6}$

This option also includes a square root sign, which is not necessary in this case.

Conclusion

Based on our analysis, the correct answer is Option A: $8\left|a^3\right|$. This option is equivalent to the simplified expression, and it includes the absolute value sign, which is necessary to ensure that the expression is valid for all values of $a$.

Final Answer

The final answer is Option A: $8\left|a^3\right|$.

Additional Tips and Resources

• To simplify radical expressions, try to factor out perfect squares and use the properties of radicals.
• When working with absolute value signs, remember that the absolute value of a number is always non-negative.
• For more practice problems and resources, check out the following websites:
  • Khan Academy: Radical Expressions and Equations
  • Mathway: Radical Expressions and Equations
  • IXL: Radical Expressions and Equations

Common Mistakes to Avoid

• Failing to factor out perfect squares when simplifying radical expressions.
• Not using the properties of radicals correctly.
• Ignoring the absolute value sign when working with radical expressions.

Real-World Applications

Radical expressions have many real-world applications, including:

• Physics: Radical expressions are used to describe the motion of objects and the behavior of waves.
• Engineering: Radical expressions are used to design and analyze complex systems, such as bridges and buildings.
• Computer Science: Radical expressions are used in algorithms and data structures to solve problems efficiently.

Conclusion

Introduction

In our previous article, we explored the process of simplifying radical expressions, with a focus on the given expression $\sqrt{64 a^6}$. We examined each option and determined which one is equivalent to the given expression. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a root or a radical sign. The most common radical sign is the square root sign, denoted by $\sqrt{}$. Radical expressions can be simplified by factoring out perfect squares or using the properties of radicals.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, try to factor out perfect squares and use the properties of radicals. You can start by breaking down the number into its prime factors and then use the property of radicals that states $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the square of an integer, such as 4 or 9. A non-perfect square is a number that cannot be expressed as the square of an integer, such as 2 or 3.

Q: How do I determine if a number is a perfect square?

A: To determine if a number is a perfect square, try to find the square root of the number. If the square root is an integer, then the number is a perfect square.

Q: What is the property of radicals that states $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$?

A: This property states that the square root of a product of two numbers is equal to the product of the square roots of the two numbers.

Q: How do I use the property of radicals to simplify a radical expression?

A: To use the property of radicals to simplify a radical expression, try to break down the number into its prime factors and then use the property to simplify the expression.

Q: What is the absolute value sign, and why is it important in radical expressions?

A: The absolute value sign is a mathematical symbol that represents the distance of a number from zero, without considering its direction. In radical expressions, the absolute value sign is important because it ensures that the expression is valid for all values of the variable.

Q: How do I simplify a radical expression with an absolute value sign?

A: To simplify a radical expression with an absolute value sign, try to factor out perfect squares and use the properties of radicals. Remember to consider the absolute value sign when working with radical expressions.

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

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

• Failing to factor out perfect squares
• Not using the properties of radicals correctly
• Ignoring the absolute value sign when working with radical expressions

Q: How do I apply radical expressions in real-world situations?

A: Radical expressions have many real-world applications, including:

• Physics: Radical expressions are used to describe the motion of objects and the behavior of waves.
• Engineering: Radical expressions are used to design and analyze complex systems, such as bridges and buildings.
• Computer Science: Radical expressions are used in algorithms and data structures to solve problems efficiently.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill to master in mathematics. By following the steps outlined in this article and using the Q&A guide, you can simplify radical expressions and apply them in real-world situations. Remember to factor out perfect squares, use the properties of radicals, and consider the absolute value sign when working with radical expressions.