Which Of The Following Is Equivalent To 10 S T ⋅ 15 T U \sqrt{10st} \cdot \sqrt{15tu} 10 S T ​ ⋅ 15 T U ​ ?A. 5 T 6 S U 5t\sqrt{6su} 5 T 6 S U ​

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Understanding the Problem

When dealing with square roots, it's essential to understand the properties of radicals and how they interact with each other. In this problem, we're given the expression 10st⋅15tu\sqrt{10st} \cdot \sqrt{15tu} and asked to find an equivalent expression.

Properties of Radicals

To solve this problem, we need to recall the properties of radicals, specifically the property that states a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}. This property allows us to combine the square roots of two numbers into a single square root of their product.

Applying the Property

Using the property mentioned above, we can rewrite the given expression as 10st⋅15tu=10st⋅15tu\sqrt{10st} \cdot \sqrt{15tu} = \sqrt{10st \cdot 15tu}.

Simplifying the Expression

Now, we can simplify the expression inside the square root by multiplying the numbers together. We have 10st⋅15tu=150st2u210st \cdot 15tu = 150st^2u^2.

Factoring the Expression

Next, we can factor the expression inside the square root to make it easier to work with. We can factor out the greatest common factor (GCF) of the expression, which is 150su2150su^2. Factoring out the GCF, we get 150su2⋅st\sqrt{150su^2 \cdot st}.

Simplifying Further

Now, we can simplify the expression further by combining the terms inside the square root. We have 150su2⋅st=150s2u2t2\sqrt{150su^2 \cdot st} = \sqrt{150s^2u^2t^2}.

Final Simplification

Finally, we can simplify the expression by taking the square root of the terms inside the square root. We have 150s2u2t2=150⋅s2⋅u2⋅t2\sqrt{150s^2u^2t^2} = \sqrt{150} \cdot \sqrt{s^2} \cdot \sqrt{u^2} \cdot \sqrt{t^2}.

Using the Property of Square Roots

Using the property of square roots that states a2=a\sqrt{a^2} = a, we can simplify the expression further. We have s2=s\sqrt{s^2} = s and u2=u\sqrt{u^2} = u and t2=t\sqrt{t^2} = t.

Final Answer

Substituting the simplified expressions back into the original expression, we get 150⋅s⋅u⋅t\sqrt{150} \cdot s \cdot u \cdot t. Now, we can simplify the expression by evaluating the square root of 150. We have 150=25⋅6=56\sqrt{150} = \sqrt{25 \cdot 6} = 5\sqrt{6}.

Final Answer

Therefore, the final answer is 56⋅stu=5st6u5\sqrt{6} \cdot stu = \boxed{5st\sqrt{6u}}.

Conclusion

In conclusion, the expression 10st⋅15tu\sqrt{10st} \cdot \sqrt{15tu} is equivalent to 5st6u5st\sqrt{6u}. This problem demonstrates the importance of understanding the properties of radicals and how they interact with each other.

Discussion

This problem is a great example of how to use the properties of radicals to simplify expressions. It's essential to remember that the square root of a product is equal to the product of the square roots. This property can be used to simplify complex expressions and make them easier to work with.

Final Thoughts

In conclusion, the expression 10st⋅15tu\sqrt{10st} \cdot \sqrt{15tu} is equivalent to 5st6u5st\sqrt{6u}. This problem demonstrates the importance of understanding the properties of radicals and how they interact with each other. By using the properties of radicals, we can simplify complex expressions and make them easier to work with.

Additional Resources

For more information on the properties of radicals, check out the following resources:

Frequently Asked Questions

Q: What is the property of radicals that allows us to combine the square roots of two numbers into a single square root of their product?

A: The property of radicals that allows us to combine the square roots of two numbers into a single square root of their product is a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}.

Q: How do we simplify the expression 10st⋅15tu\sqrt{10st} \cdot \sqrt{15tu} using the property of radicals?

A: To simplify the expression 10st⋅15tu\sqrt{10st} \cdot \sqrt{15tu}, we can use the property of radicals to combine the square roots of the two numbers into a single square root of their product. We have 10st⋅15tu=10st⋅15tu\sqrt{10st} \cdot \sqrt{15tu} = \sqrt{10st \cdot 15tu}.

Q: What is the greatest common factor (GCF) of the expression 10st⋅15tu10st \cdot 15tu?

A: The greatest common factor (GCF) of the expression 10st⋅15tu10st \cdot 15tu is 150su2150su^2.

Q: How do we simplify the expression 150su2⋅st\sqrt{150su^2 \cdot st} further?

A: To simplify the expression 150su2⋅st\sqrt{150su^2 \cdot st} further, we can combine the terms inside the square root. We have 150su2⋅st=150s2u2t2\sqrt{150su^2 \cdot st} = \sqrt{150s^2u^2t^2}.

Q: What is the final simplified expression for 150s2u2t2\sqrt{150s^2u^2t^2}?

A: The final simplified expression for 150s2u2t2\sqrt{150s^2u^2t^2} is 150⋅s2⋅u2⋅t2\sqrt{150} \cdot \sqrt{s^2} \cdot \sqrt{u^2} \cdot \sqrt{t^2}.

Q: How do we simplify the expression 150⋅s2⋅u2⋅t2\sqrt{150} \cdot \sqrt{s^2} \cdot \sqrt{u^2} \cdot \sqrt{t^2} further?

A: To simplify the expression 150⋅s2⋅u2⋅t2\sqrt{150} \cdot \sqrt{s^2} \cdot \sqrt{u^2} \cdot \sqrt{t^2} further, we can use the property of square roots that states a2=a\sqrt{a^2} = a. We have s2=s\sqrt{s^2} = s, u2=u\sqrt{u^2} = u, and t2=t\sqrt{t^2} = t.

Q: What is the final simplified expression for 150⋅s⋅u⋅t\sqrt{150} \cdot s \cdot u \cdot t?

A: The final simplified expression for 150⋅s⋅u⋅t\sqrt{150} \cdot s \cdot u \cdot t is 56⋅stu=5st6u5\sqrt{6} \cdot stu = 5st\sqrt{6u}.

Q: What is the equivalent expression for 10st⋅15tu\sqrt{10st} \cdot \sqrt{15tu}?

A: The equivalent expression for 10st⋅15tu\sqrt{10st} \cdot \sqrt{15tu} is 5st6u5st\sqrt{6u}.

Q: Why is it essential to understand the properties of radicals?

A: It's essential to understand the properties of radicals because they allow us to simplify complex expressions and make them easier to work with.

Q: What are some additional resources for learning more about the properties of radicals?

A: Some additional resources for learning more about the properties of radicals include: