What is the simplified form of the following expression?
[tex]$2 \sqrt{18}+3 \sqrt{2}+\sqrt{162}$[/tex]
Answer (2)
The simplified form of the expression 2 18 + 3 2 + 162 is 18 2 . This was achieved by simplifying each square root term and then combining like terms. Overall, the process involved breaking down the square roots into simpler components and summing the coefficients of the like terms.
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Simplify 18 to 3 2 .
Simplify 162 to 9 2 .
Substitute the simplified terms into the original expression: 2 ( 3 2 ) + 3 2 + 9 2 .
Combine like terms to get the final answer: 18 2 .
Explanation
Understanding the problem We are asked to simplify the expression 2 18 + 3 2 + 162 . To do this, we need to simplify each square root term by factoring out perfect squares.
Simplifying 18 First, let's simplify 18 . We can rewrite 18 as 9 ⋅ 2 , so 18 = 9 ⋅ 2 = 9 ⋅ 2 = 3 2 .
Simplifying 162 Next, let's simplify 162 . We can rewrite 162 as 81 ⋅ 2 , so 162 = 81 ⋅ 2 = 81 ⋅ 2 = 9 2 .
Substituting back into the expression Now, substitute the simplified square roots back into the original expression: 2 ( 3 2 ) + 3 2 + 9 2 .
Combining like terms Next, we combine like terms. 2 ( 3 2 ) + 3 2 + 9 2 = 6 2 + 3 2 + 9 2 = ( 6 + 3 + 9 ) 2 .
Calculating the final sum Finally, we calculate the sum: ( 6 + 3 + 9 ) 2 = 18 2 . Therefore, the simplified form of the expression is 18 2 .
Examples
Square roots appear in many areas of mathematics and physics. For example, when calculating the distance between two points in a coordinate plane, we often use the distance formula, which involves square roots. Simplifying expressions with square roots allows us to more easily work with these calculations and understand the relationships between different quantities. Imagine you're designing a garden and need to calculate the length of a diagonal path across a rectangular plot. Using simplified square roots helps you determine the exact length without relying on approximations.
