Simplify $\frac{\sqrt{36}}{\sqrt{3}}$.
Answer (2)
Simplify the numerator: 36 = 6 .
Rationalize the denominator: 3 6 × 3 3 = 3 6 3 .
Simplify the fraction: 3 6 3 = 2 3 .
The simplified expression is 2 3 .
Explanation
Understanding the problem We are asked to simplify the expression 3 36 . This involves simplifying a fraction with square roots.
Simplifying the numerator First, we simplify the square root in the numerator. Since 36 = 6 2 , we have 36 = 6 . So the expression becomes 3 6 .
Rationalizing the denominator To rationalize the denominator, we multiply both the numerator and the denominator by 3 . This gives us: 3 6 × 3 3 = 3 6 3 .
Simplifying the fraction Now we simplify the fraction by dividing both the numerator and the denominator by 3: 3 6 3 = 2 3 .
Final answer Therefore, the simplified expression is 2 3 .
Examples
Square roots appear in many contexts, such as calculating distances or areas. For example, if you want to find the length of the diagonal of a square with side length a , you would use the Pythagorean theorem, which involves square roots. If d is the length of the diagonal, then d 2 = a 2 + a 2 = 2 a 2 , so d = 2 a 2 = a 2 . Simplifying expressions with square roots is a fundamental skill in various mathematical and practical applications.
The simplification of 3 36 results in 2 3 . This is achieved by first simplifying the numerator and then rationalizing the denominator. Finally, we simplify the resulting fraction to obtain the answer.
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