Find the simplified product $2 \sqrt{5 x^3}\left(-3 \sqrt{10 x^2}\right)$
Answer (2)
To simplify the product 2 5 x 3 ( − 3 10 x 2 ) , we multiply the constants and the square root terms, obtaining − 30 x 2 2 x after simplification. The final answer is − 30 x 2 2 x .
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Multiply the constants: $2
\times -3 = -6$.
Multiply the square root terms: $\sqrt{5x^3}
\times \sqrt{10x^2} = \sqrt{50x^5}$.
Simplify the square root: 50 x 5 = 5 x 2 2 x .
Combine the results: $-6
\times 5x^2\sqrt{2x} = -30x^2\sqrt{2x} . T h es im pl i f i e d p ro d u c t i s \boxed{{-30 x^2 \sqrt{2 x}}}$
Explanation
Understanding the Problem We are given the expression 2 5 x 3 ( − 3 10 x 2 ) and we want to simplify it. We assume that 0"> x > 0 .
Multiplying Constants First, let's multiply the constants outside the square roots: 2 × − 3 = − 6 .
Multiplying Square Root Terms Next, let's multiply the terms inside the square roots: 5 x 3 × 10 x 2 = 5 x 3 × 10 x 2 = 50 x 5 .
Combining Results Now, combine the results from the previous steps: − 6 50 x 5 .
Simplifying the Square Root We can simplify the square root term further. We have 50 x 5 = 25 × 2 × x 4 × x = 25 × x 4 × 2 x = 5 x 2 2 x .
Final Simplification Finally, multiply the constant term with the simplified square root term: − 6 × 5 x 2 2 x = − 30 x 2 2 x .
Examples
Simplifying radical expressions is useful in various fields, such as physics and engineering, when dealing with calculations involving lengths, areas, or volumes. For example, when calculating the distance between two points in a coordinate system or determining the area of a shape with irregular sides, simplifying radical expressions can make the calculations more manageable and provide a more understandable result. Imagine you are designing a garden and need to calculate the length of a diagonal path across a rectangular plot. The length might initially be expressed as a radical, and simplifying it helps you determine the actual length for practical construction.
