Find square root of 1082.41 by prime factorisation method .
that is 108241 * 10^-2
= 49 * 2209 *10^-2
= 7*7 * 47*47 * 10^-2
square root = 7 * 47 * 10^-1
square root = 329 * 10^-1 = 32.9
To find the square root of 1082.41 by prime factorization method, follow these steps:
Step 1: Prime Factorization of 1082.41
First, find the prime factorization of 1082.41. To do this, start dividing the number by the smallest prime number, which is 2, giving you:
1082.41 ÷ 2 = 541.205
Since 541.205 is not a whole number, divide the number by the next smallest prime number, which is 3:
541.205 ÷ 3 = 180.40167
Again, since 180.40167 is not a whole number, try dividing it by the next smallest prime number, which is 5:
180.40167 ÷ 5 = 36.080334
Keep on dividing the number by prime numbers until you get a whole number.
Step 2: Simplify the Square Root Expression
Now that we have the prime factorization of 1082.41 as:
2 × 3 × 5 × 36.080334
We can simplify the square root expression by taking the square root of each prime factor.
√(1082.41) = √(2 × 3 × 5 × 36.080334)
Simplifying further, we get:
√(1082.41) = √(2) × √(3) × √(5) × √(36.080334)
Step 3: Calculate the Square Root of Each Prime Factor
Now, calculate the square root of each prime factor individually.
√(2) ≈ 1.41421356
√(3) ≈ 1.73205081
√(5) ≈ 2.23606798
√(36.080334) ≈ 6.00469331
Step 4: Combine the Square Roots
Finally, multiply the square roots of each prime factor together.
√(1082.41) ≈ 1.41421356 × 1.73205081 × 2.23606798 × 6.00469331
Calculating this expression will give you the approximate value of the square root of 1082.41 using the prime factorization method.
To find the square root of 1082.41 by prime factorization method, follow these steps:
Step 1: Prime Factorization of 1082.41
To find the square root using the prime factorization method, we need to express the number as a product of its prime factors.
1082.41 is not a perfect square, so we can start by finding the prime factorization. However, since 1082.41 is a decimal number with two decimal places, we need to move the decimal point two places to the right to work with the whole number.
1082.41 = 108241
To factorize 108241, we can divide it by prime numbers starting from 2 until we can no longer divide evenly:
Dividing by 2, we get: 108241 ÷ 2 = 54120.5 (not an integer)
Dividing by 3, we get: 108241 ÷ 3 = 36080.333 (not an integer)
Dividing by 5, we get: 108241 ÷ 5 = 21648.2 (not an integer)
Dividing by 7, we get: 108241 ÷ 7 = 15462.999 (not an integer)
Dividing by 11, we get: 108241 ÷ 11 = 9840.09 (not an integer)
Dividing by 13, we get: 108241 ÷ 13 = 8326.23 (not an integer)
Dividing by 17, we get: 108241 ÷ 17 = 6367.7 (not an integer)
Dividing by 19, we get: 108241 ÷ 19 = 5696.37 (not an integer)
Dividing by 23, we get: 108241 ÷ 23 = 4710.478 (not an integer)
At this point, we can see that no other prime numbers can divide evenly into 108241, so we can't factorize it any further.
The prime factorization of 108241 is 108241 = 7 × 11 × 13 × 17 × 19 × 23
Step 2: Grouping
Now, we will group the prime factors in pairs. Since the number under the square root is not a perfect square, any prime factor that occurs at an odd power will stay outside the square root.
Pairing the prime factors:
√108241 = √(7 × 11 × 13 × 17 × 19 × 23)
= √(7 × 11 × 17 × 23) × √(13 × 19)
= √(7 × 11 × 17 × 23) × √(13 × 19)
Step 3: Simplifying
Now, we simplify the square root of the factors that are perfect squares and keep the factors whose square roots are not integers outside the square root.
√(7 × 11 × 17 × 23) × √(13 × 19)
= √(7 × 11 × 17 × 23) × √(13 × 19)
= √707 × √437
= √707 × √(19 × 23 × 13)
= √707 × √2977
Step 4: Evaluating
Now, we evaluate the square root of the perfect squares and multiply them with the remaining square roots.
√707 × √2977
≈ 26.62 × √2977
Therefore, the approximate square root of 1082.41 by prime factorization method is 26.62
1082.41 = 108241 ÷ 100
and
108241 = 7*7*47*47
See what you can conclude from that.