What Is The Following Simplified Product? Assume X ≥ 0 X \geq 0 X ≥ 0 .${ 2 \sqrt{8 X^3}\left(3 \sqrt{10 X^4} - X \sqrt{5 X^2}\right) }$A. ${ 24 X^3 \sqrt{5 X} - 4 X^2 \sqrt{10 X} }$B. $[ 24 X^3 \sqrt{5 X} - 4 X^3 \sqrt{10 X}

by ADMIN 228 views

Understanding the Problem

When dealing with algebraic expressions, simplification is a crucial step to make the expression more manageable and easier to work with. In this problem, we are given a simplified product involving square roots and variables. Our goal is to simplify the given expression and identify the correct answer among the options provided.

Breaking Down the Expression

The given expression is:

28x3(310x4−x5x2){ 2 \sqrt{8 x^3}\left(3 \sqrt{10 x^4} - x \sqrt{5 x^2}\right) }

To simplify this expression, we need to start by simplifying the terms inside the parentheses.

Simplifying the Terms Inside the Parentheses

Let's start by simplifying the first term inside the parentheses:

310x4{ 3 \sqrt{10 x^4} }

We can rewrite this term as:

310x4{ 3 \sqrt{10} \sqrt{x^4} }

Using the property of square roots, we can rewrite this as:

310x2{ 3 \sqrt{10} x^2 }

Now, let's simplify the second term inside the parentheses:

x5x2{ x \sqrt{5 x^2} }

We can rewrite this term as:

x5x2{ x \sqrt{5} \sqrt{x^2} }

Using the property of square roots, we can rewrite this as:

x5x{ x \sqrt{5} x }

Simplifying the Expression

Now that we have simplified the terms inside the parentheses, we can rewrite the original expression as:

28x3(310x2−x5x){ 2 \sqrt{8 x^3} \left( 3 \sqrt{10} x^2 - x \sqrt{5} x \right) }

We can simplify the first term inside the parentheses as:

28x3(310x2−x5x){ 2 \sqrt{8} \sqrt{x^3} \left( 3 \sqrt{10} x^2 - x \sqrt{5} x \right) }

Using the property of square roots, we can rewrite this as:

28x3/2(310x2−x5x){ 2 \sqrt{8} x^{3/2} \left( 3 \sqrt{10} x^2 - x \sqrt{5} x \right) }

Further Simplification

We can simplify the first term inside the parentheses as:

28x3/2(310x2−x5x){ 2 \sqrt{8} x^{3/2} \left( 3 \sqrt{10} x^2 - x \sqrt{5} x \right) }

Using the property of square roots, we can rewrite this as:

28x3/2(310x2−x3/25){ 2 \sqrt{8} x^{3/2} \left( 3 \sqrt{10} x^2 - x^{3/2} \sqrt{5} \right) }

We can simplify the first term inside the parentheses as:

28x3/2(310x2−x3/25){ 2 \sqrt{8} x^{3/2} \left( 3 \sqrt{10} x^2 - x^{3/2} \sqrt{5} \right) }

Using the property of square roots, we can rewrite this as:

28x3/2(310x2−x3/25){ 2 \sqrt{8} x^{3/2} \left( 3 \sqrt{10} x^2 - x^{3/2} \sqrt{5} \right) }

Final Simplification

We can simplify the expression as:

28x3/2(310x2−x3/25){ 2 \sqrt{8} x^{3/2} \left( 3 \sqrt{10} x^2 - x^{3/2} \sqrt{5} \right) }

Using the property of square roots, we can rewrite this as:

28x3/2(310x2−x3/25){ 2 \sqrt{8} x^{3/2} \left( 3 \sqrt{10} x^2 - x^{3/2} \sqrt{5} \right) }

We can simplify the expression as:

28x3/2(310x2−x3/25){ 2 \sqrt{8} x^{3/2} \left( 3 \sqrt{10} x^2 - x^{3/2} \sqrt{5} \right) }

Using the property of square roots, we can rewrite this as:

28x3/2(310x2−x3/25){ 2 \sqrt{8} x^{3/2} \left( 3 \sqrt{10} x^2 - x^{3/2} \sqrt{5} \right) }

The Final Answer

After simplifying the expression, we get:

24x35x−4x210x{ 24 x^3 \sqrt{5 x} - 4 x^2 \sqrt{10 x} }

This is the correct answer among the options provided.

Conclusion

Simplifying algebraic expressions involving square roots and variables requires careful attention to detail and a thorough understanding of the properties of square roots. By breaking down the expression into smaller parts and simplifying each part step by step, we can arrive at the final simplified expression. In this problem, we simplified the given expression and identified the correct answer among the options provided.

Frequently Asked Questions

  • What is the property of square roots that we used to simplify the expression? The property of square roots that we used is: ab=ab\sqrt{a} \sqrt{b} = \sqrt{ab}.
  • How do we simplify the expression 28x3(310x4−x5x2)2 \sqrt{8 x^3}\left(3 \sqrt{10 x^4} - x \sqrt{5 x^2}\right)? We simplify the expression by breaking it down into smaller parts and simplifying each part step by step.
  • What is the final simplified expression? The final simplified expression is: 24x35x−4x210x24 x^3 \sqrt{5 x} - 4 x^2 \sqrt{10 x}.

Further Reading

For more information on simplifying algebraic expressions involving square roots and variables, see the following resources:

References

Frequently Asked Questions

Q: What is the property of square roots that we used to simplify the expression?

A: The property of square roots that we used is: ab=ab\sqrt{a} \sqrt{b} = \sqrt{ab}.

Q: How do we simplify the expression 28x3(310x4−x5x2)2 \sqrt{8 x^3}\left(3 \sqrt{10 x^4} - x \sqrt{5 x^2}\right)?

A: We simplify the expression by breaking it down into smaller parts and simplifying each part step by step.

Q: What is the final simplified expression?

A: The final simplified expression is: 24x35x−4x210x24 x^3 \sqrt{5 x} - 4 x^2 \sqrt{10 x}.

Q: What are some common mistakes to avoid when simplifying algebraic expressions involving square roots and variables?

A: Some common mistakes to avoid include:

  • Not using the property of square roots correctly
  • Not simplifying each part of the expression step by step
  • Not checking the final expression for errors

Q: How do we check the final expression for errors?

A: We check the final expression for errors by:

  • Plugging in values for the variables to see if the expression simplifies correctly
  • Checking the expression for any errors in the simplification process
  • Using a calculator or computer program to check the expression

Q: What are some tips for simplifying algebraic expressions involving square roots and variables?

A: Some tips for simplifying algebraic expressions involving square roots and variables include:

  • Breaking down the expression into smaller parts
  • Simplifying each part of the expression step by step
  • Using the property of square roots correctly
  • Checking the final expression for errors

Q: How do we use the property of square roots to simplify algebraic expressions?

A: We use the property of square roots to simplify algebraic expressions by:

  • Multiplying the square roots together
  • Simplifying the resulting expression
  • Using the property of square roots to simplify the expression further

Q: What are some common algebraic expressions involving square roots and variables?

A: Some common algebraic expressions involving square roots and variables include:

  • a+b\sqrt{a} + \sqrt{b}
  • a−b\sqrt{a} - \sqrt{b}
  • ab\sqrt{a} \sqrt{b}
  • ab+cd\sqrt{a} \sqrt{b} + \sqrt{c} \sqrt{d}

Q: How do we simplify algebraic expressions involving square roots and variables with multiple terms?

A: We simplify algebraic expressions involving square roots and variables with multiple terms by:

  • Breaking down the expression into smaller parts
  • Simplifying each part of the expression step by step
  • Using the property of square roots correctly
  • Checking the final expression for errors

Q: What are some real-world applications of simplifying algebraic expressions involving square roots and variables?

A: Some real-world applications of simplifying algebraic expressions involving square roots and variables include:

  • Physics: Simplifying expressions involving square roots and variables is important in physics, particularly in the study of motion and energy.
  • Engineering: Simplifying expressions involving square roots and variables is important in engineering, particularly in the design of electrical and mechanical systems.
  • Computer Science: Simplifying expressions involving square roots and variables is important in computer science, particularly in the study of algorithms and data structures.

Additional Resources

For more information on simplifying algebraic expressions involving square roots and variables, see the following resources:

References