Find the simplified product where [tex]$x \geq 0: \sqrt{5 x}\left(\sqrt{8 x^2}-2 \sqrt{x}\right)$[/tex]
Answer (2)
Distribute 5 x to both terms inside the parenthesis: 5 x 8 x 2 − 2 5 x x .
Simplify each term using the property a b = ab : 40 x 3 − 2 5 x 2 .
Simplify the radicals: 2 x 10 x − 2 x 5 .
Factor out the common factor 2 x : 2 x ( 10 x − 5 ) . The simplified expression is 2 x ( 10 x − 5 ) .
Explanation
Understanding the Problem We are given the expression 5 x ( 8 x 2 − 2 x ) where x ≥ 0 , and we want to simplify it.
Distributing the Term First, distribute the 5 x term to both terms inside the parentheses: 5 x ⋅ 8 x 2 − 5 x ⋅ 2 x = 5 x ⋅ 8 x 2 − 2 5 x ⋅ x = 40 x 3 − 2 5 x 2
Simplifying the Radicals Now, simplify the radicals. Since x ≥ 0 , we know that x 2 = x . Thus, 40 x 3 = 4 ⋅ 10 ⋅ x 2 ⋅ x = 4 x 2 ⋅ 10 x = 4 x 2 ⋅ 10 x = 2 x 10 x and 2 5 x 2 = 2 5 ⋅ x 2 = 2 x 5
Combining Terms Substitute these simplified terms back into the expression: 2 x 10 x − 2 x 5
Factoring Finally, factor out the common factor 2 x :
2 x ( 10 x − 5 ) Thus, the simplified expression is 2 x ( 10 x − 5 ) .
Final Answer The simplified product is 2 x ( 10 x − 5 ) .
Examples
This type of simplification is useful in physics when dealing with energy equations involving square roots. For example, if you are calculating kinetic energy or potential energy, you might encounter expressions with square roots that need simplification to analyze the system more effectively. Simplifying such expressions allows for easier calculations and a better understanding of the relationships between different physical quantities.
The expression 5 x ( 8 x 2 − 2 x ) simplifies to 2 x ( 10 x − 5 ) after distributing and combining like terms. The steps include distributing the radical, simplifying each term, and factoring out common elements. This process allows us to maintain clarity in simplifying expressions involving square roots.
;
