Simplify The Expression: $\[ \frac{e^3 \sqrt{f} \times E^5 F^{\frac{1}{4}}}{\left(e^{\frac{1}{5}} F^{\frac{3}{4}}\right)^{10}} \\]

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve complex problems and understand the underlying concepts. One of the most common expressions that require simplification is the one involving exponents and radicals. In this article, we will focus on simplifying the expression $\frac{e^3 \sqrt{f} \times e^5 f^{\frac{1}{4}}}{\left(e^{\frac{1}{5}} f^{\frac{3}{4}}\right)^{10}}$. We will break down the expression into smaller parts, apply the rules of exponents and radicals, and finally simplify the expression to its simplest form.

Understanding the Expression

The given expression involves exponents and radicals, which can be simplified using the rules of exponents and radicals. The expression is $\frac{e^3 \sqrt{f} \times e^5 f^{\frac{1}{4}}}{\left(e^{\frac{1}{5}} f^{\frac{3}{4}}\right)^{10}}$. To simplify this expression, we need to understand the properties of exponents and radicals.

Properties of Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $e^3$ means $e \times e \times e$. The properties of exponents are as follows:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, $e^3 \times e^5 = e^{3+5} = e^8$.
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, $(e^3)^5 = e^{3 \times 5} = e^{15}$.
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents. For example, $\frac{e^3}{e^5} = e^{3-5} = e^{-2}$.

Properties of Radicals

Radicals are a shorthand way of writing the square root of a number. For example, $\sqrt{f}$ means the square root of $f$. The properties of radicals are as follows:

• Product of Radicals Rule: When multiplying two radicals, multiply the numbers inside the radicals. For example, $\sqrt{f} \times \sqrt{g} = \sqrt{f \times g}$.
• Quotient of Radicals Rule: When dividing two radicals, divide the numbers inside the radicals. For example, $\frac{\sqrt{f}}{\sqrt{g}} = \sqrt{\frac{f}{g}}$.
• Power of a Radical Rule: When raising a radical to another power, raise the number inside the radical to that power. For example, $(\sqrt{f})^3 = f^{\frac{3}{2}}$.

Simplifying the Expression

Now that we have understood the properties of exponents and radicals, we can simplify the given expression. The expression is $\frac{e^3 \sqrt{f} \times e^5 f^{\frac{1}{4}}}{\left(e^{\frac{1}{5}} f^{\frac{3}{4}}\right)^{10}}$. To simplify this expression, we need to apply the rules of exponents and radicals.

Step 1: Simplify the Numerator

The numerator of the expression is $e^3 \sqrt{f} \times e^5 f^{\frac{1}{4}}$. We can simplify this expression using the product of powers rule and the product of radicals rule.

• Product of Powers Rule: $e^3 \times e^5 = e^{3+5} = e^8$
• Product of Radicals Rule: $\sqrt{f} \times f^{\frac{1}{4}} = \sqrt{f \times f^{\frac{1}{4}}} = \sqrt{f^{\frac{5}{4}}}$

So, the numerator becomes $e^8 \sqrt{f^{\frac{5}{4}}}$.

Step 2: Simplify the Denominator

The denominator of the expression is $\left(e^{\frac{1}{5}} f^{\frac{3}{4}}\right)^{10}$. We can simplify this expression using the power of a power rule.

• Power of a Power Rule: $\left(e^{\frac{1}{5}} f^{\frac{3}{4}}\right)^{10} = e^{\frac{1}{5} \times 10} f^{\frac{3}{4} \times 10} = e^2 f^{\frac{15}{4}}$

So, the denominator becomes $e^2 f^{\frac{15}{4}}$.

Step 3: Simplify the Expression

Now that we have simplified the numerator and the denominator, we can simplify the expression. The expression becomes $\frac{e^8 \sqrt{f^{\frac{5}{4}}}}{e^2 f^{\frac{15}{4}}}$. We can simplify this expression using the quotient of powers rule and the quotient of radicals rule.

• Quotient of Powers Rule: $\frac{e^8}{e^2} = e^{8-2} = e^6$
• Quotient of Radicals Rule: $\frac{\sqrt{f^{\frac{5}{4}}}}{f^{\frac{15}{4}}} = \sqrt{\frac{f^{\frac{5}{4}}}{f^{\frac{15}{4}}}} = \sqrt{\frac{1}{f^{\frac{5}{4}}}} = \frac{1}{\sqrt{f^{\frac{5}{4}}}} = \frac{1}{f^{\frac{5}{8}}}$

So, the expression becomes $e^6 \frac{1}{f^{\frac{5}{8}}}$.

Conclusion

In this article, we simplified the expression $\frac{e^3 \sqrt{f} \times e^5 f^{\frac{1}{4}}}{\left(e^{\frac{1}{5}} f^{\frac{3}{4}}\right)^{10}}$ using the rules of exponents and radicals. We broke down the expression into smaller parts, applied the rules of exponents and radicals, and finally simplified the expression to its simplest form. The simplified expression is $e^6 \frac{1}{f^{\frac{5}{8}}}$. This expression can be further simplified by rewriting it as $\frac{e^6}{f^{\frac{5}{8}}}$.

Introduction

In our previous article, we simplified the expression $\frac{e^3 \sqrt{f} \times e^5 f^{\frac{1}{4}}}{\left(e^{\frac{1}{5}} f^{\frac{3}{4}}\right)^{10}}$ using the rules of exponents and radicals. In this article, we will answer some of the most frequently asked questions related to simplifying expressions.

Q&A

Q: What are the properties of exponents?

A: The properties of exponents are as follows:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, $e^3 \times e^5 = e^{3+5} = e^8$.
• Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, $(e^3)^5 = e^{3 \times 5} = e^{15}$.
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents. For example, $\frac{e^3}{e^5} = e^{3-5} = e^{-2}$.

Q: What are the properties of radicals?

A: The properties of radicals are as follows:

• Product of Radicals Rule: When multiplying two radicals, multiply the numbers inside the radicals. For example, $\sqrt{f} \times \sqrt{g} = \sqrt{f \times g}$.
• Quotient of Radicals Rule: When dividing two radicals, divide the numbers inside the radicals. For example, $\frac{\sqrt{f}}{\sqrt{g}} = \sqrt{\frac{f}{g}}$.
• Power of a Radical Rule: When raising a radical to another power, raise the number inside the radical to that power. For example, $(\sqrt{f})^3 = f^{\frac{3}{2}}$.

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

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

1. Simplify the numerator: Use the product of powers rule and the product of radicals rule to simplify the numerator.
2. Simplify the denominator: Use the power of a power rule to simplify the denominator.
3. Simplify the expression: Use the quotient of powers rule and the quotient of radicals rule to simplify the expression.

Q: What is the difference between a power and a radical?

A: A power is a shorthand way of writing repeated multiplication. For example, $e^3$ means $e \times e \times e$. A radical is a shorthand way of writing the square root of a number. For example, $\sqrt{f}$ means the square root of $f$.

Q: How do I rewrite an expression with a radical in the denominator?

A: To rewrite an expression with a radical in the denominator, follow these steps:

1. Multiply the numerator and denominator by the radical: This will eliminate the radical in the denominator.
2. Simplify the expression: Use the quotient of powers rule and the quotient of radicals rule to simplify the expression.

Conclusion

In this article, we answered some of the most frequently asked questions related to simplifying expressions. We covered the properties of exponents and radicals, and provided step-by-step instructions on how to simplify an expression with exponents and radicals. We also discussed the difference between a power and a radical, and provided tips on how to rewrite an expression with a radical in the denominator.