$[ \begin Array}{|c|c|} \hline \text{Simplify The Following & (-4 \sqrt 5})(-5 \sqrt{5}) \ \hline \begin{array}{l} \text{Enter Your Most Simplified Answer As } A \sqrt{B} \ \text{If You Get An Integer Value Such As 24, Enter It As 24

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Understanding the Problem

When dealing with expressions involving square roots, it's essential to understand the properties of radicals and how they interact with multiplication. In this case, we're given the expression (−45)(−55)(-4 \sqrt{5})(-5 \sqrt{5}) and are asked to simplify it. To approach this problem, we need to recall the rules for multiplying radicals, which state that when multiplying two radicals, we can combine them by multiplying the numbers inside the radicals and then simplifying the result.

Applying the Rules of Radicals

To simplify the given expression, we can start by applying the rule for multiplying radicals. This rule states that aâ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}. We can use this rule to combine the two radicals in the given expression.

(-4 \sqrt{5})(-5 \sqrt{5}) = \sqrt{(-4)(-5)} \cdot \sqrt{5} \cdot \sqrt{5}

Simplifying the Expression

Now that we have combined the radicals, we can simplify the expression further. We can start by evaluating the product inside the first radical.

\sqrt{(-4)(-5)} = \sqrt{20}

Simplifying the Radical

Next, we can simplify the radical 20\sqrt{20} by factoring the number inside the radical. We can write 2020 as the product of 44 and 55, which are both perfect squares.

\sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2 \sqrt{5}

Combining the Simplified Radicals

Now that we have simplified the radical 20\sqrt{20}, we can combine it with the remaining radicals in the expression.

(-4 \sqrt{5})(-5 \sqrt{5}) = (2 \sqrt{5}) \cdot \sqrt{5} \cdot \sqrt{5} = 2 \sqrt{5} \cdot 5 = 10 \sqrt{5}

Conclusion

In conclusion, the simplified expression is 10510 \sqrt{5}. This is the final answer to the problem.

Discussion

This problem requires a good understanding of the rules for multiplying radicals and how to simplify expressions involving square roots. It's essential to be able to apply these rules correctly to arrive at the final answer. Additionally, this problem demonstrates the importance of factoring numbers inside radicals to simplify them.

Additional Examples

Here are a few additional examples of simplifying expressions involving radicals:

  • 16â‹…9=16â‹…9=144=12\sqrt{16} \cdot \sqrt{9} = \sqrt{16 \cdot 9} = \sqrt{144} = 12
  • (−32)(−22)=(−3)(−2)â‹…2â‹…2=6â‹…2=26(-3 \sqrt{2})(-2 \sqrt{2}) = \sqrt{(-3)(-2)} \cdot \sqrt{2} \cdot \sqrt{2} = \sqrt{6} \cdot 2 = 2 \sqrt{6}
  • 25â‹…9=25â‹…9=225=15\sqrt{25} \cdot \sqrt{9} = \sqrt{25 \cdot 9} = \sqrt{225} = 15

These examples demonstrate the importance of understanding the rules for multiplying radicals and how to simplify expressions involving square roots.

Final Thoughts

Simplifying expressions involving radicals requires a good understanding of the rules for multiplying radicals and how to simplify expressions involving square roots. By applying these rules correctly, we can arrive at the final answer to the problem. Additionally, this problem demonstrates the importance of factoring numbers inside radicals to simplify them. With practice and experience, we can become more confident in our ability to simplify expressions involving radicals.

Understanding the Problem

When dealing with expressions involving square roots, it's essential to understand the properties of radicals and how they interact with multiplication. In this case, we're given the expression (−45)(−55)(-4 \sqrt{5})(-5 \sqrt{5}) and are asked to simplify it. To approach this problem, we need to recall the rules for multiplying radicals, which state that when multiplying two radicals, we can combine them by multiplying the numbers inside the radicals and then simplifying the result.

Q&A

Q: What is the rule for multiplying radicals?

A: The rule for multiplying radicals states that when multiplying two radicals, we can combine them by multiplying the numbers inside the radicals and then simplifying the result. This can be expressed as aâ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}.

Q: How do I simplify the expression (−45)(−55)(-4 \sqrt{5})(-5 \sqrt{5})?

A: To simplify the expression (−45)(−55)(-4 \sqrt{5})(-5 \sqrt{5}), we can start by applying the rule for multiplying radicals. This rule states that a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}. We can use this rule to combine the two radicals in the given expression.

(-4 \sqrt{5})(-5 \sqrt{5}) = \sqrt{(-4)(-5)} \cdot \sqrt{5} \cdot \sqrt{5}

Q: How do I simplify the radical 20\sqrt{20}?

A: To simplify the radical 20\sqrt{20}, we can factor the number inside the radical. We can write 2020 as the product of 44 and 55, which are both perfect squares.

\sqrt{20} = \sqrt{4 \cdot 5} = \sqrt{4} \cdot \sqrt{5} = 2 \sqrt{5}

Q: What is the final answer to the problem?

A: The final answer to the problem is 10510 \sqrt{5}.

Q: Can you provide more examples of simplifying expressions involving radicals?

A: Here are a few additional examples of simplifying expressions involving radicals:

  • 16â‹…9=16â‹…9=144=12\sqrt{16} \cdot \sqrt{9} = \sqrt{16 \cdot 9} = \sqrt{144} = 12
  • (−32)(−22)=(−3)(−2)â‹…2â‹…2=6â‹…2=26(-3 \sqrt{2})(-2 \sqrt{2}) = \sqrt{(-3)(-2)} \cdot \sqrt{2} \cdot \sqrt{2} = \sqrt{6} \cdot 2 = 2 \sqrt{6}
  • 25â‹…9=25â‹…9=225=15\sqrt{25} \cdot \sqrt{9} = \sqrt{25 \cdot 9} = \sqrt{225} = 15

Conclusion

In conclusion, simplifying expressions involving radicals requires a good understanding of the rules for multiplying radicals and how to simplify expressions involving square roots. By applying these rules correctly, we can arrive at the final answer to the problem. Additionally, this problem demonstrates the importance of factoring numbers inside radicals to simplify them. With practice and experience, we can become more confident in our ability to simplify expressions involving radicals.

Final Thoughts

Simplifying expressions involving radicals is an essential skill in mathematics, and it requires a good understanding of the rules for multiplying radicals and how to simplify expressions involving square roots. By applying these rules correctly, we can arrive at the final answer to the problem. Additionally, this problem demonstrates the importance of factoring numbers inside radicals to simplify them. With practice and experience, we can become more confident in our ability to simplify expressions involving radicals.

Additional Resources