Find the simplified product where [tex]x \geq 0: \sqrt{5 x}\left(\sqrt{8 x^2}-2 \sqrt{x}\right)[/tex]
A. [tex]\sqrt{10 x}[/tex]
B. [tex]2 x \sqrt{40 x}-2 x[/tex]
C. [tex]2 x \sqrt{10 x}-2 \sqrt{5 x}[/tex]
D. [tex]2 x \sqrt{10 x}-2 x \sqrt{5}[/tex]
Answer (2)
The simplified product is 2 x 10 x − 2 x 5 , which corresponds to option D. We achieved this by distributing and simplifying each term step by step. Thus, the correct answer is option D.
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Distribute the 5 x across the terms inside the parenthesis: 5 x ⋅ 8 x 2 − 5 x ⋅ 2 x .
Simplify the first term: 5 x ⋅ 8 x 2 = 40 x 3 = 2 x 10 x .
Simplify the second term: 5 x ⋅ 2 x = 2 5 x 2 = 2 x 5 .
Combine the terms to get the final simplified expression: 2 x 10 x − 2 x 5 .
Explanation
Understanding the Problem We are given the expression 5 x ( 8 x 2 − 2 x ) and the condition x ≥ 0 . Our goal is to simplify this expression.
Distributing the terms First, distribute 5 x to both terms inside the parenthesis: 5 x ⋅ 8 x 2 − 5 x ⋅ 2 x
Simplifying the first term Now, simplify the first term: 5 x ⋅ 8 x 2 = 40 x 3 = 4 ⋅ 10 ⋅ x 2 ⋅ x = 4 x 2 ⋅ 10 x = 2 x 10 x
Simplifying the second term Next, simplify the second term: 5 x ⋅ 2 x = 2 5 x 2 = 2 x 2 ⋅ 5 = 2 x 5
Combining the terms and final answer Combine the simplified terms: 2 x 10 x − 2 x 5 Thus, the final simplified expression is 2 x 10 x − 2 x 5 .
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 might encounter expressions that require simplification using the techniques demonstrated in this problem. By simplifying these expressions, you can more easily analyze and understand the underlying phenomena.
