Simplify. [tex]$\sqrt{50}+5 \sqrt{72}$[/tex]
Answer (1)
Simplify 50 to 5 2 .
Simplify 72 to 6 2 .
Substitute the simplified square roots into the original expression: 5 2 + 5 ( 6 2 ) .
Simplify the expression to get the final answer: 35 2 .
Explanation
Understanding the Problem We are asked to simplify the expression 50 + 5 72 . To do this, we need to simplify the square roots and combine like terms.
Simplifying 50 First, let's simplify 50 . We look for the largest perfect square that divides 50. Since 50 = 25 × 2 , we have 50 = 25 × 2 = 25 × 2 = 5 2 .
Simplifying 72 Next, let's simplify 72 . We look for the largest perfect square that divides 72. Since 72 = 36 × 2 , we have 72 = 36 × 2 = 36 × 2 = 6 2 .
Substituting Back into the Expression Now, we substitute the simplified square roots back into the original expression: 50 + 5 72 = 5 2 + 5 ( 6 2 ) .
Simplifying the Expression Finally, we simplify the expression: 5 2 + 5 ( 6 2 ) = 5 2 + 30 2 = ( 5 + 30 ) 2 = 35 2 .
Examples
Square roots appear in many contexts, such as calculating distances using the Pythagorean theorem. For example, if you have a right triangle with legs of length 50 and 5 72 , the length of the hypotenuse would involve simplifying expressions with square roots. Simplifying such expressions allows for easier calculations and a better understanding of the relationships between different quantities.
