Simplify.
$\begin{array}{l}
\sqrt{80}+\sqrt{20}-\sqrt{18} \\
6 \sqrt{[?]}-\square \sqrt{\square}
\end{array}$
Answer (2)
Simplify 80 to 4 5 .
Simplify 20 to 2 5 .
Simplify 18 to 3 2 .
Combine like terms to get the final answer: 6 5 − 3 2 .
The simplified expression is 6 5 − 3 2
Explanation
Understanding the Problem We are asked to simplify the expression 80 + 20 − 18 and express it in the form 6 [ ?] − □ □ . Let's break down each term to simplify the expression.
Simplifying 80 First, we simplify 80 . We look for the largest perfect square that divides 80. Since 80 = 16 × 5 , we have 80 = 16 × 5 = 16 × 5 = 4 5 .
Simplifying 20 Next, we simplify 20 . We look for the largest perfect square that divides 20. Since 20 = 4 × 5 , we have 20 = 4 × 5 = 4 × 5 = 2 5 .
Simplifying 18 Now, we simplify 18 . We look for the largest perfect square that divides 18. Since 18 = 9 × 2 , we have 18 = 9 × 2 = 9 × 2 = 3 2 .
Substituting Back Now we substitute these simplified expressions back into the original expression: 80 + 20 − 18 = 4 5 + 2 5 − 3 2 .
Combining Like Terms We combine the like terms, which are the terms with 5 :
4 5 + 2 5 = ( 4 + 2 ) 5 = 6 5 . So the expression becomes 6 5 − 3 2 .
Final Answer Finally, we express the simplified expression in the requested format: 6 [ ?] − □ □ . Comparing 6 5 − 3 2 with 6 [ ?] − □ □ , we can see that [?] = 5, the first square is 3, and the second square is 2.
Examples
Square roots are used in many fields, including engineering, physics, and computer science. For example, when calculating the distance between two points in a coordinate plane, we use the distance formula, which involves square roots. Simplifying radical expressions makes these calculations easier and more accurate. Imagine you're designing a bridge and need to calculate the length of support cables. Using simplified square roots ensures your measurements are precise, preventing structural issues.
The expression 80 + 20 − 18 simplifies to 6 5 − 3 2 . This result was obtained by simplifying each square root individually and then combining like terms. The final format matches the requested form with values being 5 and 2 for the squares.
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