Simplify $7 \sqrt{3}+\sqrt{12}-\sqrt{75}$
Answer (2)
Simplify s q r t 12 to 2 s q r t 3 .
Simplify s q r t 75 to 5 s q r t 3 .
Substitute the simplified terms into the original expression: 7 s q r t 3 + 2 s q r t 3 − 5 s q r t 3 .
Combine like terms to get the final answer: b o x e d 4 s q r t 3 .
Explanation
Understanding the problem We are asked to simplify the expression 7 s q r t 3 + s q r t 12 − s q r t 75 . This involves simplifying the square roots and combining like terms.
Simplifying s q r t 12 First, we simplify s q r t 12 . We can rewrite 12 as 4 t im es 3 , so s q r t 12 = s q r t 4 t im es 3 = s q r t 4 t im es s q r t 3 = 2 s q r t 3 .
Simplifying s q r t 75 Next, we simplify s q r t 75 . We can rewrite 75 as 25 t im es 3 , so s q r t 75 = s q r t 25 t im es 3 = s q r t 25 t im es s q r t 3 = 5 s q r t 3 .
Substituting back into the expression Now we substitute the simplified square roots back into the original expression: 7 s q r t 3 + 2 s q r t 3 − 5 s q r t 3 .
Combining like terms We combine the terms by adding and subtracting the coefficients of s q r t 3 : ( 7 + 2 − 5 ) s q r t 3 = ( 9 − 5 ) s q r t 3 = 4 s q r t 3 .
Final Answer Therefore, the simplified expression is 4 s q r t 3 .
Examples
Square roots appear in many contexts, including geometry and physics. For example, when calculating the length of the diagonal of a square with side length s , the diagonal is s s q r t 2 . Simplifying expressions with square roots is a fundamental skill in these areas. Imagine you are building a square garden and need to calculate the length of a support beam that runs diagonally across the garden. If the side of the garden is s q r t 8 meters, the diagonal would be s q r t 8 t im es s q r t 2 = s q r t 16 = 4 meters. Simplifying radical expressions allows for easier and more accurate calculations in such practical scenarios.
To simplify the expression 7 3 + 12 − 75 , we first simplify the square roots to get 2 3 and 5 3 , respectively. Combining these terms leads us to a final answer of 4 3 .
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