Multiply. Assume P P P And Q Q Q Are Greater Than Or Equal To Zero, And Write Your Answer In Simplest Form. 21 P 3 Q 2 ⋅ 5 P 2 Q 2 \sqrt{21 P^3 Q^2} \cdot \sqrt{5 P^2 Q^2} 21 P 3 Q 2 ​ ⋅ 5 P 2 Q 2 ​

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Understanding the Problem

When dealing with radical expressions, it's essential to understand the properties of radicals and how to simplify them. In this problem, we're given two radical expressions, 21p3q2\sqrt{21 p^3 q^2} and 5p2q2\sqrt{5 p^2 q^2}, and we're asked to multiply them together. To simplify the resulting expression, we'll use the properties of radicals and exponents.

Properties of Radicals

Before we dive into the problem, let's review some essential properties of radicals:

  • Product Rule: a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}
  • Power Rule: ab=ab/2\sqrt{a^b} = a^{b/2}
  • Zero Exponent Rule: a0=1a^0 = 1

Simplifying the Radical Expressions

Now that we've reviewed the properties of radicals, let's simplify the given radical expressions:

21p3q2⋅5p2q2\sqrt{21 p^3 q^2} \cdot \sqrt{5 p^2 q^2}

Using the Product Rule, we can combine the two radical expressions into a single radical expression:

21p3q2⋅5p2q2\sqrt{21 p^3 q^2 \cdot 5 p^2 q^2}

Next, we can simplify the expression inside the radical by multiplying the numbers and combining the variables:

105p5q4\sqrt{105 p^5 q^4}

Simplifying the Expression Inside the Radical

Now that we have the expression inside the radical simplified, let's focus on simplifying the radical expression itself. We can use the Power Rule to simplify the expression:

105p5q4=105⋅p5⋅q4\sqrt{105 p^5 q^4} = \sqrt{105} \cdot \sqrt{p^5} \cdot \sqrt{q^4}

Using the Power Rule, we can rewrite the expression as:

105⋅p5/2⋅q4/2\sqrt{105} \cdot p^{5/2} \cdot q^{4/2}

Simplifying further, we get:

105⋅p5/2⋅q2\sqrt{105} \cdot p^{5/2} \cdot q^2

Final Answer

The final answer is:

105⋅p5/2⋅q2\boxed{\sqrt{105} \cdot p^{5/2} \cdot q^2}

Conclusion

In this problem, we used the properties of radicals and exponents to simplify the given radical expressions. By applying the Product Rule, Power Rule, and Zero Exponent Rule, we were able to simplify the expression inside the radical and ultimately arrive at the final answer. This problem demonstrates the importance of understanding the properties of radicals and how to apply them to simplify complex expressions.

Example Use Cases

Radical expressions are used in various mathematical applications, including:

  • Algebra: Radical expressions are used to simplify and solve equations.
  • Geometry: Radical expressions are used to calculate lengths and areas of shapes.
  • Trigonometry: Radical expressions are used to simplify and solve trigonometric equations.

Tips and Tricks

When working with radical expressions, it's essential to remember the following tips and tricks:

  • Use the Product Rule: When multiplying two radical expressions, use the Product Rule to combine them into a single radical expression.
  • Use the Power Rule: When simplifying a radical expression, use the Power Rule to rewrite the expression in a simpler form.
  • Simplify the expression inside the radical: Before simplifying the radical expression itself, simplify the expression inside the radical by multiplying the numbers and combining the variables.

Frequently Asked Questions

In this article, we'll address some common questions and concerns related to simplifying radical expressions. Whether you're a student, teacher, or simply looking to brush up on your math skills, this Q&A section is designed to provide you with the answers you need.

Q: What is the Product Rule for radicals?

A: The Product Rule for radicals states that a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}. This means that when multiplying two radical expressions, you can combine them into a single radical expression by multiplying the numbers and variables inside the radicals.

Q: How do I apply the Power Rule for radicals?

A: The Power Rule for radicals states that ab=ab/2\sqrt{a^b} = a^{b/2}. To apply this rule, simply rewrite the expression inside the radical as a power of the variable, and then simplify the resulting expression.

Q: What is the Zero Exponent Rule for radicals?

A: The Zero Exponent Rule for radicals states that a0=1a^0 = 1. This means that any number or variable raised to the power of zero is equal to 1.

Q: How do I simplify a radical expression with multiple variables?

A: To simplify a radical expression with multiple variables, simply multiply the numbers and variables inside the radicals, and then simplify the resulting expression using the Power Rule and other properties of radicals.

Q: Can I simplify a radical expression with a negative exponent?

A: Yes, you can simplify a radical expression with a negative exponent by rewriting the expression as a fraction and then simplifying the resulting expression.

Q: How do I apply the Product Rule for radicals to simplify a complex expression?

A: To apply the Product Rule for radicals to simplify a complex expression, simply multiply the numbers and variables inside the radicals, and then simplify the resulting expression using the Power Rule and other properties of radicals.

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

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

  • Not simplifying the expression inside the radical: Make sure to simplify the expression inside the radical before simplifying the radical expression itself.
  • Not using the Power Rule: Make sure to use the Power Rule to rewrite the expression inside the radical as a power of the variable.
  • Not simplifying the resulting expression: Make sure to simplify the resulting expression after applying the Power Rule and other properties of radicals.

Q: How do I check my work when simplifying radical expressions?

A: To check your work when simplifying radical expressions, simply plug the simplified expression back into the original equation and verify that it is true. You can also use a calculator or other tool to check your work.

Conclusion

In this Q&A article, we've addressed some common questions and concerns related to simplifying radical expressions. By following the tips and tricks outlined in this article, you'll be able to simplify radical expressions with ease and arrive at the final answer with confidence.

Example Use Cases

Radical expressions are used in various mathematical applications, including:

  • Algebra: Radical expressions are used to simplify and solve equations.
  • Geometry: Radical expressions are used to calculate lengths and areas of shapes.
  • Trigonometry: Radical expressions are used to simplify and solve trigonometric equations.

Tips and Tricks

When working with radical expressions, it's essential to remember the following tips and tricks:

  • Use the Product Rule: When multiplying two radical expressions, use the Product Rule to combine them into a single radical expression.
  • Use the Power Rule: When simplifying a radical expression, use the Power Rule to rewrite the expression in a simpler form.
  • Simplify the expression inside the radical: Before simplifying the radical expression itself, simplify the expression inside the radical by multiplying the numbers and combining the variables.

By following these tips and tricks, you'll be able to simplify radical expressions with ease and arrive at the final answer with confidence.