What is the following product?
[tex]$\sqrt{30} \cdot \sqrt{10}$[/tex]
Answer (2)
The product 30 ⋅ 10 simplifies to 10 3 by using the property of square roots. First, we combine the square roots to get 300 , then factor and simplify. Thus, the final answer is 10 3 .
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Use the property a ⋅ b = a ⋅ b to rewrite the expression as 30 ⋅ 10 = 300 .
Factor 300 as 100 ⋅ 3 , so 300 = 100 ⋅ 3 .
Use the property a ⋅ b = a ⋅ b to rewrite the expression as 100 ⋅ 3 .
Simplify 100 to 10, so the final answer is 10 3 .
Explanation
Understanding the Problem We are asked to find the product of two square roots: 30 ⋅ 10 . We need to simplify this expression and choose the correct answer from the given options.
Using Properties of Square Roots To simplify the product of square roots, we can use the property that a ⋅ b = a ⋅ b . Applying this property, we get: 30 ⋅ 10 = 30 ⋅ 10 = 300
Factoring the Number Inside the Square Root Now, we need to simplify 300 . We can factor 300 as 300 = 100 ⋅ 3 . Therefore, we have: 300 = 100 ⋅ 3
Separating the Square Root Using the property a ⋅ b = a ⋅ b again, we can rewrite the expression as: 100 ⋅ 3 = 100 ⋅ 3
Simplifying the Square Root Since 100 = 10 , we have: 100 ⋅ 3 = 10 ⋅ 3 = 10 3
Choosing the Correct Answer Comparing our simplified expression 10 3 with the given options, we see that it matches the last option.
Final Answer Therefore, the product 30 ⋅ 10 simplifies to 10 3 .
Examples
Understanding how to simplify square roots is useful in many areas, such as physics and engineering, when dealing with distances or areas. For example, if you are calculating the length of the diagonal of a rectangle with sides of length 30 and 10 , you would need to calculate ( 30 ) 2 + ( 10 ) 2 = 30 + 10 = 40 = 2 10 . Simplifying radicals helps in expressing these quantities in their simplest form.
