Perform The Operations And Simplify: $\frac{\sqrt{5 X^7}}{\sqrt{3 X}} \cdot \sqrt{15 X^2}$A. $5 X^4$ B. $\sqrt{25 X^3}$ C. $25 X^8$ D. $5 X^2 \sqrt{x^6}$

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Introduction

Radical expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore the process of simplifying radical expressions, with a focus on the given problem: 5x73xâ‹…15x2\frac{\sqrt{5 x^7}}{\sqrt{3 x}} \cdot \sqrt{15 x^2}. We will break down the solution into manageable steps, using the properties of radicals and exponents to simplify the expression.

Understanding Radical Expressions

Before we dive into the solution, let's take a moment to understand what radical expressions are. A radical expression is any expression that contains a square root or other root symbol. The most common radical expression is the square root, denoted by the symbol \sqrt{}. For example, 4\sqrt{4} is a radical expression because it contains a square root.

Properties of Radicals

Radicals have several properties that make them easier to work with. Here are a few key properties:

  • Product Property: aâ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}
  • Quotient Property: ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}
  • Power Property: ab=ab2\sqrt{a^b} = a^{\frac{b}{2}}

Simplifying the Given Expression

Now that we have a good understanding of radical expressions and their properties, let's simplify the given expression: 5x73xâ‹…15x2\frac{\sqrt{5 x^7}}{\sqrt{3 x}} \cdot \sqrt{15 x^2}.

Step 1: Simplify the Square Roots

To simplify the expression, we can start by simplifying the square roots. We can rewrite the expression as:

5x73xâ‹…15x2=5â‹…x73â‹…xâ‹…15x2\frac{\sqrt{5 x^7}}{\sqrt{3 x}} \cdot \sqrt{15 x^2} = \frac{\sqrt{5} \cdot \sqrt{x^7}}{\sqrt{3} \cdot \sqrt{x}} \cdot \sqrt{15 x^2}

Step 2: Apply the Product Property

Next, we can apply the product property to simplify the expression further:

5â‹…x73â‹…xâ‹…15x2=5â‹…x723â‹…x12â‹…15x2\frac{\sqrt{5} \cdot \sqrt{x^7}}{\sqrt{3} \cdot \sqrt{x}} \cdot \sqrt{15 x^2} = \frac{\sqrt{5} \cdot x^{\frac{7}{2}}}{\sqrt{3} \cdot x^{\frac{1}{2}}} \cdot \sqrt{15 x^2}

Step 3: Apply the Quotient Property

Now, we can apply the quotient property to simplify the expression further:

5⋅x723⋅x12⋅15x2=53⋅x72−12⋅15x2\frac{\sqrt{5} \cdot x^{\frac{7}{2}}}{\sqrt{3} \cdot x^{\frac{1}{2}}} \cdot \sqrt{15 x^2} = \sqrt{\frac{5}{3}} \cdot x^{\frac{7}{2} - \frac{1}{2}} \cdot \sqrt{15 x^2}

Step 4: Simplify the Exponents

Next, we can simplify the exponents:

53⋅x72−12⋅15x2=53⋅x3⋅15x2\sqrt{\frac{5}{3}} \cdot x^{\frac{7}{2} - \frac{1}{2}} \cdot \sqrt{15 x^2} = \sqrt{\frac{5}{3}} \cdot x^3 \cdot \sqrt{15 x^2}

Step 5: Apply the Product Property Again

Finally, we can apply the product property again to simplify the expression further:

53â‹…x3â‹…15x2=53â‹…15â‹…x6\sqrt{\frac{5}{3}} \cdot x^3 \cdot \sqrt{15 x^2} = \sqrt{\frac{5}{3} \cdot 15 \cdot x^6}

Step 6: Simplify the Expression

Now, we can simplify the expression:

53â‹…15â‹…x6=753â‹…x6=25â‹…x6=5x3x3\sqrt{\frac{5}{3} \cdot 15 \cdot x^6} = \sqrt{\frac{75}{3} \cdot x^6} = \sqrt{25 \cdot x^6} = 5 x^3 \sqrt{x^3}

Conclusion

In conclusion, the simplified expression is 5x3x35 x^3 \sqrt{x^3}. This is the final answer to the given problem.

Final Answer

The final answer is: 5x3x3\boxed{5 x^3 \sqrt{x^3}}

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Introduction

In our previous article, we explored the process of simplifying radical expressions, with a focus on the given problem: 5x73xâ‹…15x2\frac{\sqrt{5 x^7}}{\sqrt{3 x}} \cdot \sqrt{15 x^2}. We broke down the solution into manageable steps, using the properties of radicals and exponents to simplify the expression. In this article, we will answer some common questions related to simplifying radical expressions.

Q&A

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

A: A radical expression is any expression that contains a square root or other root symbol, while an exponential expression is any expression that contains a power or exponent. For example, 4\sqrt{4} is a radical expression, while 222^2 is an exponential expression.

Q: What are the properties of radicals?

A: Radicals have several properties that make them easier to work with. Here are a few key properties:

  • Product Property: aâ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}
  • Quotient Property: ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}
  • Power Property: ab=ab2\sqrt{a^b} = a^{\frac{b}{2}}

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can follow these steps:

  1. Simplify the square roots.
  2. Apply the product property.
  3. Apply the quotient property.
  4. Simplify the exponents.
  5. Apply the product property again.

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

A: A rational exponent is an exponent that can be expressed as a fraction, while an irrational exponent is an exponent that cannot be expressed as a fraction. For example, x12x^{\frac{1}{2}} is a rational exponent, while x2x^{\sqrt{2}} is an irrational exponent.

Q: How do I simplify a radical expression with a rational exponent?

A: To simplify a radical expression with a rational exponent, you can follow these steps:

  1. Simplify the square roots.
  2. Apply the product property.
  3. Apply the quotient property.
  4. Simplify the exponents.
  5. Apply the product property again.

Q: What is the difference between a radical expression and a rational expression?

A: A radical expression is any expression that contains a square root or other root symbol, while a rational expression is any expression that contains a fraction. For example, 4\sqrt{4} is a radical expression, while 23\frac{2}{3} is a rational expression.

Q: How do I simplify a radical expression with a rational expression?

A: To simplify a radical expression with a rational expression, you can follow these steps:

  1. Simplify the square roots.
  2. Apply the product property.
  3. Apply the quotient property.
  4. Simplify the exponents.
  5. Apply the product property again.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for any math enthusiast. By understanding the properties of radicals and following the steps outlined in this article, you can simplify even the most complex radical expressions. Remember to always simplify the square roots, apply the product property, apply the quotient property, simplify the exponents, and apply the product property again.

Final Tips

  • Always simplify the square roots first.
  • Apply the product property and quotient property in that order.
  • Simplify the exponents before applying the product property again.
  • Practice, practice, practice! The more you practice simplifying radical expressions, the more comfortable you will become with the process.

Final Answer

The final answer is: 5x3x3\boxed{5 x^3 \sqrt{x^3}}