Find The Simplified Product: 2 \sqrt{5 X^3} \left(-3 \sqrt{10 X^2}\right ]A. − 30 2 X 5 -30 \sqrt{2 X^5} − 30 2 X 5 ​ B. − 30 X 2 2 X -30 X^2 \sqrt{2 X} − 30 X 2 2 X ​ C. − 12 X 2 5 X -12 X^2 \sqrt{5 X} − 12 X 2 5 X ​ D. − 6 50 X 5 -6 \sqrt{50 X^5} − 6 50 X 5 ​

by ADMIN 262 views

=====================================================

Understanding the Problem

When dealing with the product of radicals, it's essential to simplify the expression by combining like terms and eliminating any unnecessary radicals. In this problem, we're given the expression 25x3(−310x2)2 \sqrt{5 x^3} \left(-3 \sqrt{10 x^2}\right) and asked to find its simplified product.

The Rules of Simplifying Radicals

Before we dive into the solution, let's review the rules for simplifying radicals:

  • When multiplying two or more radicals, we can combine them by multiplying the numbers inside the radicals.
  • When multiplying a radical by a non-radical, we can simply multiply the numbers inside the radical.
  • When multiplying two or more radicals with the same index, we can combine them by multiplying the numbers inside the radicals.

Applying the Rules to the Given Expression

Now that we've reviewed the rules, let's apply them to the given expression:

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

To simplify this expression, we'll start by multiplying the numbers inside the radicals:

25x3(−310x2)=2⋅−35x3⋅10x22 \sqrt{5 x^3} \left(-3 \sqrt{10 x^2}\right) = 2 \cdot -3 \sqrt{5 x^3 \cdot 10 x^2}

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

2⋅−35x3⋅10x2=−650x52 \cdot -3 \sqrt{5 x^3 \cdot 10 x^2} = -6 \sqrt{50 x^5}

Simplifying the Radical

Now that we've simplified the expression inside the radical, let's simplify the radical itself:

−650x5-6 \sqrt{50 x^5}

To simplify the radical, we'll look for any perfect squares that can be factored out of the expression inside the radical. In this case, we can factor out a perfect square of 25:

−650x5=−625⋅2x5-6 \sqrt{50 x^5} = -6 \sqrt{25 \cdot 2 x^5}

Next, we'll simplify the expression inside the radical by taking the square root of the perfect square:

−625⋅2x5=−6⋅52x5-6 \sqrt{25 \cdot 2 x^5} = -6 \cdot 5 \sqrt{2 x^5}

Final Answer

Now that we've simplified the expression, let's look at the answer choices to see which one matches our solution:

A. −302x5-30 \sqrt{2 x^5} B. −30x22x-30 x^2 \sqrt{2 x} C. −12x25x-12 x^2 \sqrt{5 x} D. −650x5-6 \sqrt{50 x^5}

Our solution matches answer choice D, −650x5-6 \sqrt{50 x^5}, but we can simplify it further by factoring out a perfect square of 2:

−650x5=−62⋅25x5-6 \sqrt{50 x^5} = -6 \sqrt{2 \cdot 25 x^5}

−62⋅25x5=−6⋅52x5-6 \sqrt{2 \cdot 25 x^5} = -6 \cdot 5 \sqrt{2 x^5}

However, we can simplify it even further by factoring out a perfect square of x2x^2:

−6⋅52x5=−6⋅5⋅x22x-6 \cdot 5 \sqrt{2 x^5} = -6 \cdot 5 \cdot x^2 \sqrt{2 x}

−6⋅5⋅x22x=−30x22x-6 \cdot 5 \cdot x^2 \sqrt{2 x} = -30 x^2 \sqrt{2 x}

Therefore, the final answer is:

B. −30x22x-30 x^2 \sqrt{2 x}

Conclusion

Simplifying the product of radicals requires careful attention to the rules of simplifying radicals. By following these rules and simplifying the expression step-by-step, we can arrive at the final answer. In this case, the final answer is −30x22x-30 x^2 \sqrt{2 x}.

=====================================================

Frequently Asked Questions

Q: What are the rules for simplifying radicals?

A: When multiplying two or more radicals, we can combine them by multiplying the numbers inside the radicals. When multiplying a radical by a non-radical, we can simply multiply the numbers inside the radical. When multiplying two or more radicals with the same index, we can combine them by multiplying the numbers inside the radicals.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, start by multiplying the numbers inside the radicals. Then, simplify the expression inside the radical by multiplying the numbers and combining the variables. Finally, simplify the radical itself by looking for any perfect squares that can be factored out of the expression inside the radical.

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

A: A perfect square is a number that can be expressed as the square of an integer, such as 4 (2^2) or 9 (3^2). A perfect cube is a number that can be expressed as the cube of an integer, such as 8 (2^3) or 27 (3^3).

Q: How do I simplify a radical expression with a perfect square?

A: To simplify a radical expression with a perfect square, factor out the perfect square from the expression inside the radical. Then, take the square root of the perfect square and simplify the remaining expression.

Q: Can I simplify a radical expression with a variable?

A: Yes, you can simplify a radical expression with a variable. To do this, follow the same steps as simplifying a radical expression with a number. However, be careful when combining variables, as the order of operations may affect the final result.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is −30x22x-30 x^2 \sqrt{2 x}.

Q: Can I use a calculator to simplify a radical expression?

A: While a calculator can be a useful tool for simplifying radical expressions, it's not always the best option. Calculators may not always display the simplified form of the expression, and they may not be able to handle complex expressions. It's usually best to simplify radical expressions by hand, using the rules and techniques outlined above.

Q: How do I know if a radical expression is simplified?

A: A radical expression is simplified when it can be expressed in the simplest form possible, with no unnecessary radicals or variables. To check if a radical expression is simplified, look for any perfect squares that can be factored out of the expression inside the radical. If you can simplify the expression further, then it's not yet simplified.

Common Mistakes to Avoid

When simplifying radical expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not multiplying the numbers inside the radicals: When multiplying two or more radicals, make sure to multiply the numbers inside the radicals.
  • Not simplifying the expression inside the radical: When simplifying a radical expression, make sure to simplify the expression inside the radical by multiplying the numbers and combining the variables.
  • Not looking for perfect squares: When simplifying a radical expression, make sure to look for any perfect squares that can be factored out of the expression inside the radical.
  • Not following the order of operations: When simplifying a radical expression, make sure to follow the order of operations (PEMDAS) to ensure that the expression is simplified correctly.

Conclusion

Simplifying radical expressions requires careful attention to the rules and techniques outlined above. By following these rules and techniques, you can simplify even the most complex radical expressions. Remember to avoid common mistakes, such as not multiplying the numbers inside the radicals or not looking for perfect squares. With practice and patience, you'll become a pro at simplifying radical expressions in no time!