Simplify 3681⎯⎯⎯⎯√
. Select all that apply.
The given expression 3681√ cannot be simplified further.
To simplify the square root of 3681, we can try to find any perfect square factors.
Let's start by finding the prime factorization of 3681:
- Divide 3681 by 2: 3681 ÷ 2 = 1840.5 (not a whole number)
- Divide 3681 by 3: 3681 ÷ 3 = 1227 (whole number)
- Divide 1227 by 3: 1227 ÷ 3 = 409 (whole number)
So, the prime factorization of 3681 is 3 × 3 × 409.
Now, let's group the prime factors into pairs of the same value:
3 × 3 × 409 = (3 × 3) × 409 = 9 × 409
Therefore, the simplified square root of 3681 is √(9 × 409).
However, we can't simplify the square root of 409 further because it doesn't have any perfect square factors.
So, the simplified square root of 3681 is 3√409.
To simplify the expression √3681, there are a few approaches.
1. Prime factorization:
Find the prime factors of 3681. Divide it by prime numbers starting from 2 until you cannot divide anymore.
3681 ÷ 3 = 1227, so one prime factor is 3.
1227 ÷ 3 = 409, so another prime factor is 3.
409 is a prime number itself.
Therefore, the prime factorization of 3681 is 3 × 3 × 409.
Since we are trying to simplify the square root, we can rewrite this expression as follows:
√3681 = √(3 × 3 × 409)
Now we can simplify further using a property of square roots:
√(a × b) = √a × √b
√(3 × 3 × 409) = √3 × √3 × √409
The square root of 3 multiplied by itself ( √3 × √3) is simply 3:
√3 × √3 × √409 = 3 × √409
Therefore, the simplified expression is 3√409.
2. Factor pairs:
Find the factor pairs of 3681 and look for perfect squares.
The factor pairs of 3681 are:
1 × 3681
3 × 1227
9 × 409
From these factor pairs, we can see that 409 is a perfect square since it has only one factor pair, 9 × 409.
We can rewrite √3681 as follows:
√3681 = √(9 × 409)
Using the property of square roots mentioned earlier, we can simplify further:
√(9 × 409) = √9 × √409 = 3 × √409
Therefore, the simplified expression is 3√409.
So, the simplified expression is 3√409.