Simplify the following expression: $(\sqrt{3 x}+\sqrt{5})(\sqrt{15 x}+2 \sqrt{30})$
Answer (2)
The expression ( 3 x + 5 ) ( 15 x + 2 30 ) simplifies to 3 x 5 + 6 10 x + 5 3 x + 10 6 . This is achieved by expanding and simplifying each term appropriately. The final result combines all simplified terms for the complete expression.
;
Expand the original expression: ( 3 x + 5 ) ( 15 x + 2 30 ) = 45 x 2 + 2 90 x + 75 x + 2 150 .
Simplify each term: 45 x 2 = 3 x 5 , 2 90 x = 6 10 x , 75 x = 5 3 x , 2 150 = 10 6 .
Combine the simplified terms: 3 x 5 + 6 10 x + 5 3 x + 10 6 .
The correct simplified expression is 3 x 5 + 6 10 x + 5 3 x + 10 6 .
Explanation
Problem Analysis We are given the expression ( 3 x + 5 ) ( 15 x + 2 30 ) and four possible simplified forms. Our goal is to determine which of the simplified forms is correct. We will expand the original expression and simplify it to see which of the given options matches our result.
Expanding the Expression First, we expand the expression: ( 3 x + 5 ) ( 15 x + 2 30 ) = 3 x ⋅ 15 x + 3 x ⋅ 2 30 + 5 ⋅ 15 x + 5 ⋅ 2 30 = 45 x 2 + 2 90 x + 75 x + 2 150
Simplifying Each Term Next, we simplify each term: 45 x 2 = 9 ⋅ 5 ⋅ x 2 = 3 x 5 2 90 x = 2 9 ⋅ 10 ⋅ x = 2 ⋅ 3 10 x = 6 10 x 75 x = 25 ⋅ 3 ⋅ x = 5 3 x 2 150 = 2 25 ⋅ 6 = 2 ⋅ 5 6 = 10 6
Combining Terms Combining the simplified terms, we get: 3 x 5 + 6 10 x + 5 3 x + 10 6
Final Answer Comparing our simplified expression with the given options, we find that it matches the second option: 3 x 5 + 6 10 x + 5 3 x + 10 6
Conclusion Therefore, the correct simplified expression is 3 x 5 + 6 10 x + 5 3 x + 10 6 .
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, if you are designing a square garden and need to calculate the length of the diagonal, you might end up with an expression like 2 x 2 , which simplifies to x 2 . This skill is also essential in computer graphics for calculating distances and transformations.
