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

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Understanding the Problem

When dealing with radical expressions, it's essential to understand the rules for simplifying products. In this case, 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 for Simplifying Products

To simplify the product of two radical expressions, we need to follow these rules:

  • When multiplying two radical expressions with the same index, we can combine the radicands by multiplying them.
  • When multiplying two radical expressions with different indices, we need to find the least common multiple (LCM) of the indices and rewrite each radical expression with that index.
  • We can then combine the radicands by multiplying them.

Applying the Rules to the Given Expression

Let's apply these rules to the given expression:

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

First, we can combine the radicands by multiplying them:

5x3â‹…10x2=5â‹…10â‹…x3â‹…x2\sqrt{5 x^3} \cdot \sqrt{10 x^2} = \sqrt{5 \cdot 10 \cdot x^3 \cdot x^2}

Next, we can simplify the radicand by combining like terms:

50x5\sqrt{50 x^5}

Now, we can multiply the coefficients:

2⋅−3=−62 \cdot -3 = -6

So, the simplified product is:

−650x5-6 \sqrt{50 x^5}

Checking the Answer Choices

Let's check the answer choices to see if they match our simplified product:

A. −302x5-30 \sqrt{2 x^5}

This is not the same as our simplified product.

B. −30x22x-30 x^2 \sqrt{2 x}

This is not the same as our simplified product.

C. −12x25x-12 x^2 \sqrt{5 x}

This is not the same as our simplified product.

D. −650x5-6 \sqrt{50 x^5}

This is the same as our simplified product.

Conclusion

In this example, we applied the rules for simplifying products of radical expressions to find the simplified product of 25x3(−310x2)2 \sqrt{5 x^3} \left(-3 \sqrt{10 x^2}\right). We combined the radicands by multiplying them, simplified the radicand by combining like terms, and multiplied the coefficients. The simplified product is −650x5-6 \sqrt{50 x^5}.

Frequently Asked Questions

Q: What are the rules for simplifying products of radical expressions?

A: The rules for simplifying products of radical expressions are:

  • When multiplying two radical expressions with the same index, we can combine the radicands by multiplying them.
  • When multiplying two radical expressions with different indices, we need to find the least common multiple (LCM) of the indices and rewrite each radical expression with that index.
  • We can then combine the radicands by multiplying them.

Q: How do I simplify the product of two radical expressions?

A: To simplify the product of two radical expressions, you need to follow the rules for simplifying products of radical expressions. This involves combining the radicands by multiplying them, simplifying the radicand by combining like terms, and multiplying the coefficients.

Q: What is the simplified product of 25x3(−310x2)2 \sqrt{5 x^3} \left(-3 \sqrt{10 x^2}\right)?

A: The simplified product of 25x3(−310x2)2 \sqrt{5 x^3} \left(-3 \sqrt{10 x^2}\right) is −650x5-6 \sqrt{50 x^5}.

Additional Resources

For more information on simplifying products of radical expressions, check out the following resources:

Conclusion

In conclusion, simplifying the product of radical expressions involves applying the rules for simplifying products of radical expressions. This includes combining the radicands by multiplying them, simplifying the radicand by combining like terms, and multiplying the coefficients. By following these rules, we can simplify complex expressions and find the final answer.

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Q: What are the rules for simplifying products of radical expressions?

A: The rules for simplifying products of radical expressions are:

  • When multiplying two radical expressions with the same index, we can combine the radicands by multiplying them.
  • When multiplying two radical expressions with different indices, we need to find the least common multiple (LCM) of the indices and rewrite each radical expression with that index.
  • We can then combine the radicands by multiplying them.

Q: How do I simplify the product of two radical expressions?

A: To simplify the product of two radical expressions, you need to follow the rules for simplifying products of radical expressions. This involves combining the radicands by multiplying them, simplifying the radicand by combining like terms, and multiplying the coefficients.

Q: What is the simplified product of 25x3(−310x2)2 \sqrt{5 x^3} \left(-3 \sqrt{10 x^2}\right)?

A: The simplified product of 25x3(−310x2)2 \sqrt{5 x^3} \left(-3 \sqrt{10 x^2}\right) is −650x5-6 \sqrt{50 x^5}.

Q: How do I handle negative numbers when simplifying products of radical expressions?

A: When simplifying products of radical expressions, you need to handle negative numbers by multiplying them with the coefficient of the radical expression. For example, if you have −310x2-3 \sqrt{10 x^2}, you would multiply the negative sign with the coefficient of the radical expression, resulting in −310x2-3 \sqrt{10 x^2}.

Q: Can I simplify the product of two radical expressions with different indices?

A: Yes, you can simplify the product of two radical expressions with different indices by finding the least common multiple (LCM) of the indices and rewriting each radical expression with that index. You can then combine the radicands by multiplying them.

Q: How do I find the least common multiple (LCM) of two indices?

A: To find the least common multiple (LCM) of two indices, you need to list the multiples of each index and find the smallest multiple that is common to both. For example, if you have the indices 3 and 4, the multiples of 3 are 3, 6, 9, 12, and the multiples of 4 are 4, 8, 12. The least common multiple of 3 and 4 is 12.

Q: Can I simplify the product of two radical expressions with a variable in the radicand?

A: Yes, you can simplify the product of two radical expressions with a variable in the radicand by combining the variables and simplifying the radicand. For example, if you have 5x3â‹…10x2\sqrt{5 x^3} \cdot \sqrt{10 x^2}, you can combine the variables by multiplying them, resulting in 50x5\sqrt{50 x^5}.

Q: How do I simplify the product of two radical expressions with a coefficient in the radicand?

A: To simplify the product of two radical expressions with a coefficient in the radicand, you need to multiply the coefficients and combine the radicands. For example, if you have 25x3â‹…310x22 \sqrt{5 x^3} \cdot 3 \sqrt{10 x^2}, you would multiply the coefficients, resulting in 650x56 \sqrt{50 x^5}.

Q: Can I simplify the product of two radical expressions with a fraction in the radicand?

A: Yes, you can simplify the product of two radical expressions with a fraction in the radicand by multiplying the fractions and combining the radicands. For example, if you have 52x3â‹…103x2\sqrt{\frac{5}{2} x^3} \cdot \sqrt{\frac{10}{3} x^2}, you would multiply the fractions, resulting in 506x5\sqrt{\frac{50}{6} x^5}.

Q: How do I simplify the product of two radical expressions with a negative number in the radicand?

A: To simplify the product of two radical expressions with a negative number in the radicand, you need to multiply the negative number with the coefficient of the radical expression and combine the radicands. For example, if you have −3−5x3⋅210x2-3 \sqrt{-5 x^3} \cdot 2 \sqrt{10 x^2}, you would multiply the negative number with the coefficient of the radical expression, resulting in −650x5-6 \sqrt{50 x^5}.

Q: Can I simplify the product of two radical expressions with a complex number in the radicand?

A: Yes, you can simplify the product of two radical expressions with a complex number in the radicand by multiplying the complex numbers and combining the radicands. For example, if you have 3+4i⋅5−6i\sqrt{3 + 4i} \cdot \sqrt{5 - 6i}, you would multiply the complex numbers, resulting in 15−18i\sqrt{15 - 18i}.

Q: How do I simplify the product of two radical expressions with a radical in the radicand?

A: To simplify the product of two radical expressions with a radical in the radicand, you need to multiply the radicals and combine the radicands. For example, if you have 5x3â‹…10x2\sqrt{\sqrt{5 x^3}} \cdot \sqrt{\sqrt{10 x^2}}, you would multiply the radicals, resulting in 50x5\sqrt{50 x^5}.

Q: Can I simplify the product of two radical expressions with a power in the radicand?

A: Yes, you can simplify the product of two radical expressions with a power in the radicand by multiplying the powers and combining the radicands. For example, if you have x3â‹…x2\sqrt{x^3} \cdot \sqrt{x^2}, you would multiply the powers, resulting in x5\sqrt{x^5}.

Q: How do I simplify the product of two radical expressions with a variable in the coefficient?

A: To simplify the product of two radical expressions with a variable in the coefficient, you need to multiply the variables and combine the coefficients. For example, if you have 2x5x3â‹…3y10x22x \sqrt{5 x^3} \cdot 3y \sqrt{10 x^2}, you would multiply the variables, resulting in 6xy50x56xy \sqrt{50 x^5}.

Q: Can I simplify the product of two radical expressions with a fraction in the coefficient?

A: Yes, you can simplify the product of two radical expressions with a fraction in the coefficient by multiplying the fractions and combining the coefficients. For example, if you have 235x3â‹…3410x2\frac{2}{3} \sqrt{5 x^3} \cdot \frac{3}{4} \sqrt{10 x^2}, you would multiply the fractions, resulting in 1250x5\frac{1}{2} \sqrt{50 x^5}.

Q: How do I simplify the product of two radical expressions with a negative number in the coefficient?

A: To simplify the product of two radical expressions with a negative number in the coefficient, you need to multiply the negative number with the coefficient of the radical expression and combine the coefficients. For example, if you have −25x3⋅310x2-2 \sqrt{5 x^3} \cdot 3 \sqrt{10 x^2}, you would multiply the negative number with the coefficient of the radical expression, resulting in −650x5-6 \sqrt{50 x^5}.

Q: Can I simplify the product of two radical expressions with a complex number in the coefficient?

A: Yes, you can simplify the product of two radical expressions with a complex number in the coefficient by multiplying the complex numbers and combining the coefficients. For example, if you have (3+4i)5x3⋅(5−6i)10x2(3 + 4i) \sqrt{5 x^3} \cdot (5 - 6i) \sqrt{10 x^2}, you would multiply the complex numbers, resulting in (15−18i)50x5(15 - 18i) \sqrt{50 x^5}.

Q: How do I simplify the product of two radical expressions with a radical in the coefficient?

A: To simplify the product of two radical expressions with a radical in the coefficient, you need to multiply the radicals and combine the coefficients. For example, if you have 5x3â‹…10x2\sqrt{5 x^3} \cdot \sqrt{10 x^2}, you would multiply the radicals, resulting in 50x5\sqrt{50 x^5}.

Q: Can I simplify the product of two radical expressions with a power in the coefficient?

A: Yes, you can simplify the product of two radical expressions with a power in the coefficient by multiplying the powers and combining the coefficients. For example, if you have x25x3â‹…y310x2x^2 \sqrt{5 x^3} \cdot y^3 \sqrt{10 x^2}, you would multiply the powers, resulting in x5y350x5x^5 y^3 \sqrt{50 x^5}.

Q: How do I simplify the product of two radical expressions with a variable in the radicand and a variable in the coefficient?

A: To simplify the product of two radical expressions with a variable in the radicand and a variable in the coefficient, you need to multiply the variables and combine the radicands and coefficients. For example, if you have x5x3â‹…y10x2x \sqrt{5 x^3} \cdot y \sqrt{10 x^2}, you would multiply the variables, resulting in xy50x5xy \sqrt{50 x^5}.

Q: Can I