Find the simplified product: [tex]2 \sqrt{5 x^3}\left(-3 \sqrt{10 x^2}\right)[/tex]
Answer (2)
The simplified product of the expression 2 5 x 3 ( − 3 10 x 2 ) is − 30 x 2 2 x . This was achieved by multiplying coefficients, combining and simplifying the square root terms. The final result illustrates how to handle and simplify expressions involving square roots and variables in algebra.
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Multiply the coefficients: 2 × − 3 = − 6 .
Multiply the terms inside the square roots: 5 x 3 × 10 x 2 = 50 x 5 .
Simplify the square root: 50 x 5 = 5 x 2 2 x .
Multiply the simplified square root by the coefficient: − 6 × 5 x 2 2 x = − 30 x 2 2 x . The final simplified expression is $\boxed{{-30 x^2
\sqrt{2 x}}}$. ### Explanation 1. Understanding the Problem We are given the expression $2 \sqrt{5 x^3}(-3
10 x 2 ) and we want to simplify it.
Multiplying Coefficients First, let's multiply the coefficients outside the square roots:
2 × − 3 = − 6
Multiplying Terms Inside Square Roots 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 two steps:
− 6 50 x 5
Simplifying the Square Root Simplify the square root. We can rewrite 50 as 25 × 2 and x 5 as x 4 × x . Thus,
50 x 5 = 25 × 2 × x 4 × x = 25 x 4 × 2 x = 25 × x 4 × 2 x = 5 x 2 2 x
Final Simplification Finally, multiply the simplified square root by the coefficient:
− 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 quantities involving square roots. For example, when calculating the impedance of an electrical circuit or the energy of a quantum particle, you often encounter expressions with radicals that need to be simplified for further calculations. Simplifying these expressions makes the calculations easier and more accurate. Also, in geometry, when finding the length of a diagonal or the area of certain shapes, you might need to simplify radical expressions.
