What Is The Simplified Form Of The Following Expression? Assume $a \geq 0$ And $c \geq 0$.$\[ 14\left(\sqrt[4]{a^5 B^2 C^4}\right) - 7 A C\left(\sqrt[4]{a B^2}\right) \\]A. $\[ 7 A C\left(\sqrt[4]{a B^2}\right) \\]B.

Introduction

In this article, we will explore the simplified form of a given mathematical expression. The expression involves the use of exponents, radicals, and algebraic manipulation. We will assume that $a \geq 0$ and $c \geq 0$ to simplify the expression.

The Given Expression

The given expression is:

${ 14\left(\sqrt[4]{a^5 b^2 c^4}\right) - 7 a c\left(\sqrt[4]{a b^2}\right) }$

Step 1: Simplify the Radicals

To simplify the radicals, we will use the property of exponents that states $\sqrt[n]{x^n} = x$. We will also use the property of radicals that states $\sqrt[n]{x^m} = x^{m/n}$.

Let's simplify the first radical:

${ \sqrt[4]{a^5 b^2 c^4} = a^{5/4} b^{2/4} c^{4/4} = a^{5/4} b^{1/2} c }$

Now, let's simplify the second radical:

${ \sqrt[4]{a b^2} = a^{1/4} b^{2/4} = a^{1/4} b^{1/2} }$

Step 2: Substitute the Simplified Radicals

Now that we have simplified the radicals, we can substitute them back into the original expression:

${ 14\left(a^{5/4} b^{1/2} c\right) - 7 a c\left(a^{1/4} b^{1/2}\right) }$

Step 3: Distribute the Coefficients

To simplify the expression further, we will distribute the coefficients:

${ 14a^{5/4} b^{1/2} c - 7a^{1+1/4} b^{1/2} c }$

Step 4: Simplify the Exponents

Now that we have distributed the coefficients, we can simplify the exponents:

${ 14a^{5/4} b^{1/2} c - 7a^{5/4} b^{1/2} c }$

Step 5: Combine Like Terms

Now that we have simplified the exponents, we can combine like terms:

${ 14a^{5/4} b^{1/2} c - 7a^{5/4} b^{1/2} c = 7a^{5/4} b^{1/2} c }$

Conclusion

In conclusion, the simplified form of the given expression is:

${ 7a^{5/4} b^{1/2} c }$

This expression is the result of simplifying the original expression using the properties of exponents and radicals.

Answer

The correct answer is:

${ 7a^{5/4} b^{1/2} c }$

This answer is the result of simplifying the original expression using the properties of exponents and radicals.

Discussion

The given expression involves the use of exponents, radicals, and algebraic manipulation. To simplify the expression, we used the properties of exponents and radicals. We also used the distributive property to simplify the expression further.

The simplified form of the expression is:

${ 7a^{5/4} b^{1/2} c }$

This expression is the result of simplifying the original expression using the properties of exponents and radicals.

Final Answer

The final answer is:

${ 7a^{5/4} b^{1/2} c }$

Introduction

In our previous article, we explored the simplified form of a given mathematical expression. The expression involved the use of exponents, radicals, and algebraic manipulation. We assumed that $a \geq 0$ and $c \geq 0$ to simplify the expression.

In this article, we will answer some frequently asked questions (FAQs) related to the simplified form of the given expression.

Q: What is the simplified form of the expression?

A: The simplified form of the expression is:

${ 7a^{5/4} b^{1/2} c }$

Q: How did you simplify the expression?

A: We used the properties of exponents and radicals to simplify the expression. We also used the distributive property to simplify the expression further.

Q: What are the properties of exponents and radicals that you used?

A: We used the following properties of exponents and radicals:

• $\sqrt[n]{x^n} = x$
• $\sqrt[n]{x^m} = x^{m/n}$
• $a^{m/n} = \sqrt[n]{a^m}$

Q: Why did you assume that $a \geq 0$ and $c \geq 0$?

A: We assumed that $a \geq 0$ and $c \geq 0$ to simplify the expression. This assumption is necessary because the expression involves the use of radicals, and radicals are only defined for non-negative numbers.

Q: Can you explain the distributive property that you used?

A: The distributive property states that:

${ a(b + c) = ab + ac }$

We used this property to simplify the expression by distributing the coefficients.

Q: How can I apply the simplified form of the expression to real-world problems?

A: The simplified form of the expression can be applied to real-world problems that involve the use of exponents and radicals. For example, you can use this expression to simplify complex mathematical expressions that involve the use of exponents and radicals.

Q: What are some common mistakes that people make when simplifying expressions involving exponents and radicals?

A: Some common mistakes that people make when simplifying expressions involving exponents and radicals include:

• Not using the properties of exponents and radicals correctly
• Not distributing the coefficients correctly
• Not assuming that $a \geq 0$ and $c \geq 0$ when simplifying expressions involving radicals

Conclusion

In conclusion, the simplified form of the given expression is:

${ 7a^{5/4} b^{1/2} c }$

We used the properties of exponents and radicals to simplify the expression, and we also used the distributive property to simplify the expression further. We hope that this article has helped to clarify any questions that you may have had about the simplified form of the expression.

Final Answer

The final answer is:

${ 7a^{5/4} b^{1/2} c }$

This answer is the result of simplifying the original expression using the properties of exponents and radicals.