What is a simple way to find the radical form of a square root? i.e. the number 368
Break the number down into the highest perfect square that it divides into plus whatever number you multiply it by. Examples: sqrt (square root) of 50, you break that down into 25 (perfect square) x 2, the 25 is 5 squared so your answer is 5 sqrt 2. Example 2: sqrt 147, break down into 49 x 3, 7 sqrt 3
To find the radical form of a square root, such as the square root of 368, you can follow these steps:
Step 1: Prime factorization
Find the prime factorization of the number inside the square root, in this case, 368. To do this, divide it by the smallest prime number and continue dividing until you have only prime numbers left or you cannot divide any further.
For 368:
368 ÷ 2 = 184
184 ÷ 2 = 92
92 ÷ 2 = 46
46 ÷ 2 = 23
The prime factorization of 368 is 2 × 2 × 2 × 2 × 23, which can be written as 2^4 × 23.
Step 2: Grouping into pairs
Group the prime factors in pairs. Take the first two factors, then the next two, and so on. If there is an odd factor left at the end, leave it alone.
For the prime factors 2^4 × 23:
Group the first two 2's: 2 × 2 = 4
Leave the last two digits (one 2 and the 23) without pairing.
Step 3: Simplify the radical expression
Write the square root using the prime factors grouped in pairs and individually for any remaining factor.
Square root of 368 can be written as:
√368 = √(2^4 × 23)
Apply the grouping:
√(2^4 × 23) = √(4 × 2 × 2 × 23)
Combine the grouped factors and the individual remaining factor:
√(4 × 2 × 2 × 23) = √(4) × √(2 × 2 × 23)
Simplify each part:
√(4) × √(2 × 2 × 23) = 2 × 2√(23) = 4√(23)
Therefore, the radical form of the square root of 368 is 4√23.