What is the following sum in simplest form?
$\sqrt{8}+3 \sqrt{2}+\sqrt{32}$
Answer (1)
Simplify 8 to 2 2 .
Simplify 32 to 4 2 .
Substitute the simplified terms into the original expression: 2 2 + 3 2 + 4 2 .
Combine like terms to get the final answer: 9 2 .
Explanation
Understanding the Problem We are asked to simplify the sum 8 + 3 2 + 32 . This involves simplifying each square root term and then combining like terms.
Simplifying 8 First, we simplify 8 . We can rewrite 8 as 4 ⋅ 2 , so 8 = 4 ⋅ 2 = 4 ⋅ 2 = 2 2 .
Simplifying 32 Next, we simplify 32 . We can rewrite 32 as 16 ⋅ 2 , so 32 = 16 ⋅ 2 = 16 ⋅ 2 = 4 2 .
Substituting Simplified Terms Now, we substitute the simplified terms back into the original expression: 2 2 + 3 2 + 4 2 .
Combining Like Terms Finally, we combine like terms: ( 2 + 3 + 4 ) 2 = 9 2 .
Final Answer Therefore, the simplified sum is 9 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 distances and other related quantities. Imagine you're building a garden and need to calculate the length of a diagonal support beam. If the sides of your garden form a right triangle, the length of the beam is the square root of the sum of the squares of the sides. Simplifying this square root will give you the exact length you need to cut the beam.
