Understanding the Square Root of 147
The square root of a number is the value that, when multiplied by itself, equals the original number. In the case of 147, we want to find the number that, when squared (multiplied by itself), results in 147.
Finding the Square Root: Prime Factorization
One method to find the square root of 147 is through prime factorization. This involves breaking down the number into its prime factors:
-
Find the prime factors of 147:
- 147 is divisible by 3, giving us 3 * 49.
- 49 is divisible by 7, giving us 7 * 7.
- Therefore, the prime factorization of 147 is 3 * 7 * 7.
-
Look for pairs of prime factors: We have a pair of 7's.
-
Take one factor from each pair: For every pair of prime factors, we take one of the factors outside the square root. In this case, we take one 7 outside the square root.
-
The remaining factors stay inside the square root: The remaining factor, 3, stays inside the square root.
Therefore, the square root of 147 can be represented as: √147 = 7√3
Understanding the Result
The result 7√3 means that the square root of 147 is equal to 7 multiplied by the square root of 3. This is a simplified form, as the square root of 3 cannot be expressed as a whole number.
Approximating the Value
While the square root of 3 is irrational (cannot be expressed as a simple fraction), we can approximate its value using a calculator or a table of square roots. The approximation of √3 is approximately 1.732. Therefore, the approximate value of the square root of 147 is:
7 * 1.732 ≈ 12.124
Summary
Finding the square root of 147 involves:
- Prime factorization: Breaking down 147 into its prime factors (3, 7, 7).
- Pairing prime factors: Identifying pairs of identical prime factors (7, 7).
- Simplifying: Taking one factor from each pair outside the square root (7) and leaving the remaining factor inside (3).
This process results in the simplified form 7√3, which represents the exact square root of 147. We can approximate the value of √3 to obtain an approximate value for the square root of 147.