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 multiplication. We will assume that $a \geq 0$ and $c \geq 0$, and we will use these assumptions to simplify the expression.

Understanding the Expression

The given expression is:

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

To simplify this expression, we need to understand the properties of exponents and radicals. We will use the following properties:

• The product rule for exponents: $a^m \cdot a^n = a^{m+n}$
• The power rule for exponents: $(a^m)^n = a^{mn}$
• The property of radicals: $\sqrt[n]{a^n} = a$

Simplifying the Expression

We will start by simplifying the first term of the expression:

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

Using the property of radicals, we can rewrite this term as:

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

Now, we will simplify the second term of the expression:

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

Using the property of radicals, we can rewrite this term as:

$$7 a c\left(a^{\frac{1}{4}} b^{\frac{1}{2}}\right)$$

Combining the Terms

Now that we have simplified both terms, we can combine them:

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

We can factor out the common term $b^{\frac{1}{2}}$ from both terms:

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

Simplifying Further

We can simplify the expression inside the parentheses:

$$14 a^{\frac{5}{4}} c - 7 a c a^{\frac{1}{4}}$$

Using the product rule for exponents, we can rewrite this expression as:

$$14 a^{\frac{5}{4}} c - 7 a^{\frac{5}{4}} c$$

Final Simplification

Now, we can simplify the expression by combining the two terms:

$$14 a^{\frac{5}{4}} c - 7 a^{\frac{5}{4}} c = 7 a^{\frac{5}{4}} c$$

Conclusion

The simplified form of the given expression is:

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

This is the final answer.

Final Answer

The final answer is $\boxed{7 a c\left(\sqrt[4]{a b^2}\right)}$.

Introduction

In our previous article, we explored the simplified form of a given mathematical expression. We used various properties of exponents and radicals to simplify the expression. In this article, we will answer some frequently asked questions (FAQs) about simplifying mathematical expressions.

Q: What are the most common properties of exponents and radicals that I should know?

A: The most common properties of exponents and radicals that you should know are:

• The product rule for exponents: $a^m \cdot a^n = a^{m+n}$
• The power rule for exponents: $(a^m)^n = a^{mn}$
• The property of radicals: $\sqrt[n]{a^n} = a$
• The property of radicals: $\sqrt[n]{a} = a^{\frac{1}{n}}$

Q: How do I simplify an expression with multiple terms?

A: To simplify an expression with multiple terms, you can use the following steps:

1. Simplify each term separately using the properties of exponents and radicals.
2. Combine the simplified terms using the rules of arithmetic.
3. Factor out any common terms from the expression.

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

A: A radical is a symbol that represents the nth root of a number, while an exponent is a number that represents the power to which a number is raised. For example, $\sqrt{a}$ is a radical, while $a^2$ is an exponent.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can use the following steps:

1. Rewrite the expression with a positive exponent by taking the reciprocal of the base.
2. Simplify the expression using the properties of exponents and radicals.
3. Combine the simplified terms using the rules of arithmetic.

Q: What is the most important thing to remember when simplifying mathematical expressions?

A: The most important thing to remember when simplifying mathematical expressions is to use the properties of exponents and radicals correctly. This will help you to simplify the expression accurately and efficiently.

Q: Can I use a calculator to simplify mathematical expressions?

A: Yes, you can use a calculator to simplify mathematical expressions. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

Q: How do I know when to use the product rule for exponents versus the power rule for exponents?

A: To determine whether to use the product rule for exponents or the power rule for exponents, you can ask yourself the following questions:

• Is the expression a product of two or more terms? If so, use the product rule for exponents.
• Is the expression a power of a single term? If so, use the power rule for exponents.

Q: Can I simplify an expression with a variable in the exponent?

A: Yes, you can simplify an expression with a variable in the exponent. However, you will need to use the properties of exponents and radicals correctly to simplify the expression.

Q: How do I simplify an expression with a fraction in the exponent?

A: To simplify an expression with a fraction in the exponent, you can use the following steps:

1. Rewrite the fraction as a decimal or a percentage.
2. Simplify the expression using the properties of exponents and radicals.
3. Combine the simplified terms using the rules of arithmetic.

Conclusion

Simplifying mathematical expressions can be a challenging task, but with practice and patience, you can become proficient in simplifying expressions using the properties of exponents and radicals. Remember to use the product rule for exponents, the power rule for exponents, and the property of radicals correctly to simplify expressions.