Go step by step to reduce the radical.
√32
Step 1: Find the largest perfect square that divides 32 evenly.
32 can be written as 16 * 2. Therefore, √32 = √(16 * 2).
Step 2: Break down the square root into separate square roots.
√(16 * 2) = √16 * √2
Step 3: Simplify the square root of the perfect square.
√16 = 4
Step 4: Put it all together.
√32 = 4√2
Therefore, √32 simplifies to 4√2.
Go step by step to reduce the radical √200
Step 1: Find the largest perfect square that divides 200 evenly.
200 can be written as 100 * 2. Therefore, √200 = √(100 * 2).
Step 2: Break down the square root into separate square roots.
√(100 * 2) = √100 * √2
Step 3: Simplify the square root of the perfect square.
√100 = 10
Step 4: Put it all together.
√200 = 10√2
Therefore, √200 simplifies to 10√2.
Express in simplest radical form √72.
To simplify the square root of 72, we need to find the largest perfect square that divides evenly into 72.
Step 1: Factor 72 into its prime factors:
72 = 2 * 2 * 2 * 3 * 3 or 2^3 * 3^2
Step 2: Identify the perfect squares within the factorization. In this case, we have 2^2 = 4.
Step 3: Rewrite the square root using the perfect square factor:
√72 = √(4 * 18)
Step 4: Break down the square root into separate square roots:
√(4 * 18) = √4 * √18
Step 5: Simplify the square root of the perfect square:
√4 = 2
Step 6: Put it all together:
√72 = 2√18
Step 7: Continue simplifying the square root until no perfect squares remain:
√18 = √(9 * 2)
√18 = √9 * √2
√18 = 3√2
Therefore, the simplest radical form of √72 is 2√18 or 6√2.