Find The Product: 18 X 5 Y 2 ⋅ 72 X Y 3 \sqrt{18 X^5 Y^2} \cdot \sqrt{72 X Y^3} 18 X 5 Y 2 ​ ⋅ 72 X Y 3 ​ A. 36 X Y \sqrt{36 X Y} 36 X Y ​ B. 36 X 3 Y 2 36 \sqrt{x^3 Y^2} 36 X 3 Y 2 ​ C. 36 X 3 Y 2 36 X^3 Y^2 36 X 3 Y 2 D. 36 X 3 Y 2 Y 36 X^3 Y^2 \sqrt{y} 36 X 3 Y 2 Y ​

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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 article, we'll focus on simplifying the given expression: 18x5y2⋅72xy3\sqrt{18 x^5 y^2} \cdot \sqrt{72 x y^3}. Our goal is to find the product of these two radical expressions and simplify it to its simplest form.

Properties of Radicals

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

  • The product of two square roots is equal to the square root of the product of the two numbers inside the square roots.
  • The product of two cube roots is equal to the cube root of the product of the two numbers inside the cube roots.
  • The product of two nth roots is equal to the nth root of the product of the two numbers inside the nth roots.

Simplifying the Expression

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

18x5y2⋅72xy3\sqrt{18 x^5 y^2} \cdot \sqrt{72 x y^3}

To simplify this expression, we'll use the property that the product of two square roots is equal to the square root of the product of the two numbers inside the square roots.

18x5y2⋅72xy3=18x5y2⋅72xy3\sqrt{18 x^5 y^2} \cdot \sqrt{72 x y^3} = \sqrt{18 x^5 y^2 \cdot 72 x y^3}

Next, we'll simplify the expression inside the square root by multiplying the numbers and combining the variables.

18x5y2⋅72xy3=1296x6y5\sqrt{18 x^5 y^2 \cdot 72 x y^3} = \sqrt{1296 x^6 y^5}

Simplifying the Expression Further

Now that we have the expression inside the square root, let's simplify it further by factoring out perfect squares.

1296x6y5=(1296x6y5)\sqrt{1296 x^6 y^5} = \sqrt{(1296 x^6 y^5)}

To simplify this expression, we'll look for perfect squares that can be factored out of the expression inside the square root.

(1296x6y5)=(362⋅x6⋅y4⋅y)\sqrt{(1296 x^6 y^5)} = \sqrt{(36^2 \cdot x^6 \cdot y^4 \cdot y)}

Simplifying the Expression Even Further

Now that we have the expression inside the square root, let's simplify it even further by factoring out perfect squares.

(362⋅x6⋅y4⋅y)=(362⋅x6⋅y4)⋅y\sqrt{(36^2 \cdot x^6 \cdot y^4 \cdot y)} = \sqrt{(36^2 \cdot x^6 \cdot y^4) \cdot y}

To simplify this expression, we'll look for perfect squares that can be factored out of the expression inside the square root.

(362⋅x6⋅y4)⋅y=36x3y2y\sqrt{(36^2 \cdot x^6 \cdot y^4) \cdot y} = 36 x^3 y^2 \sqrt{y}

Conclusion

In this article, we've simplified the given expression: 18x5y2⋅72xy3\sqrt{18 x^5 y^2} \cdot \sqrt{72 x y^3}. We've used the properties of radicals to simplify the expression and factored out perfect squares to simplify it further. The final simplified expression is 36x3y2y36 x^3 y^2 \sqrt{y}.

Final Answer

The final answer is 36x3y2y36 x^3 y^2 \sqrt{y}.

Discussion

This problem requires a good understanding of the properties of radicals and how to simplify them. It's essential to remember that the product of two square roots is equal to the square root of the product of the two numbers inside the square roots. Additionally, it's crucial to factor out perfect squares to simplify the expression further.

Related Problems

If you're looking for more problems to practice, here are a few related problems:

  • Simplify the expression: 16x3y2⋅9xy4\sqrt{16 x^3 y^2} \cdot \sqrt{9 x y^4}
  • Simplify the expression: 25x2y3⋅36xy5\sqrt{25 x^2 y^3} \cdot \sqrt{36 x y^5}
  • Simplify the expression: 81x4y2⋅49x3y4\sqrt{81 x^4 y^2} \cdot \sqrt{49 x^3 y^4}

References

If you're looking for more information on simplifying radical expressions, here are a few references:

  • Khan Academy: Simplifying Radical Expressions
  • Mathway: Simplifying Radical Expressions
  • Purplemath: Simplifying Radical Expressions

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Frequently Asked Questions

Q: What is the product of two square roots?

A: The product of two square roots is equal to the square root of the product of the two numbers inside the square roots.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the properties of radicals to combine the numbers and variables inside the square root. You can also factor out perfect squares to simplify the expression further.

Q: What is the difference between a square root and a cube root?

A: A square root is a radical expression that represents the number that, when multiplied by itself, gives the original number. A cube root is a radical expression that represents the number that, when multiplied by itself twice, gives the original number.

Q: How do I simplify a cube root expression?

A: To simplify a cube root expression, you can use the properties of cube roots to combine the numbers and variables inside the cube root. You can also factor out perfect cubes to simplify the expression further.

Q: What is the product of two cube roots?

A: The product of two cube roots is equal to the cube root of the product of the two numbers inside the cube roots.

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

A: To simplify a radical expression with multiple variables, you can use the properties of radicals to combine the numbers and variables inside the square root. You can also factor out perfect squares to simplify the expression further.

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

A: A rational expression is a fraction that contains variables and/or numbers in the numerator and/or denominator. A radical expression is a mathematical expression that contains a square root or cube root.

Q: How do I simplify a rational expression with a radical in the numerator or denominator?

A: To simplify a rational expression with a radical in the numerator or denominator, you can use the properties of radicals to combine the numbers and variables inside the square root. You can also factor out perfect squares to simplify the expression further.

Common Mistakes to Avoid

Mistake 1: Not using the properties of radicals to simplify the expression

A: When simplifying a radical expression, it's essential to use the properties of radicals to combine the numbers and variables inside the square root.

Mistake 2: Not factoring out perfect squares

A: When simplifying a radical expression, it's essential to factor out perfect squares to simplify the expression further.

Mistake 3: Not using the correct order of operations

A: When simplifying a radical expression, it's essential to use the correct order of operations to simplify the expression.

Tips and Tricks

Tip 1: Use the properties of radicals to simplify the expression

A: When simplifying a radical expression, it's essential to use the properties of radicals to combine the numbers and variables inside the square root.

Tip 2: Factor out perfect squares

A: When simplifying a radical expression, it's essential to factor out perfect squares to simplify the expression further.

Tip 3: Use the correct order of operations

A: When simplifying a radical expression, it's essential to use the correct order of operations to simplify the expression.

Conclusion

In this article, we've answered some frequently asked questions about simplifying radical expressions. We've also discussed common mistakes to avoid and provided tips and tricks for simplifying radical expressions.

Final Answer

The final answer is 36x3y2y36 x^3 y^2 \sqrt{y}.

Discussion

This problem requires a good understanding of the properties of radicals and how to simplify them. It's essential to remember that the product of two square roots is equal to the square root of the product of the two numbers inside the square roots. Additionally, it's crucial to factor out perfect squares to simplify the expression further.

Related Problems

If you're looking for more problems to practice, here are a few related problems:

  • Simplify the expression: 16x3y2⋅9xy4\sqrt{16 x^3 y^2} \cdot \sqrt{9 x y^4}
  • Simplify the expression: 25x2y3⋅36xy5\sqrt{25 x^2 y^3} \cdot \sqrt{36 x y^5}
  • Simplify the expression: 81x4y2⋅49x3y4\sqrt{81 x^4 y^2} \cdot \sqrt{49 x^3 y^4}

References

If you're looking for more information on simplifying radical expressions, here are a few references:

  • Khan Academy: Simplifying Radical Expressions
  • Mathway: Simplifying Radical Expressions
  • Purplemath: Simplifying Radical Expressions