reduse the radical. ^192
The simplified form of the expression √192 is 8√3.
To simplify the square root of 192, we can break it down into its factors. The prime factorization of 192 is:
192 = 2^6 * 3
Now, we look for pairs of duplicate prime factors. In this case, we have two 2's that can be taken out from under the square root:
√192 = √(2^6 * 3)
Taking out the pairs of duplicate factors, we can simplify it as:
√192 = √(2^2 * 2^2 * 2^2 * 3)
Simplifying further,
√192 = √(2^2) * √(2^2) * √(2^2) * √3
We can then take the square root of each term:
√192 = 2 * 2 * 2 * √3
Simplifying,
√192 = 8√3
So the simplified form of √192 is 8√3.
To simplify or reduce a radical expression, we will try to find perfect square factors of the number under the radical sign (the radicand) and rewrite it as the product of the perfect square and any remaining factors. Let's simplify the radical expression √192.
Step 1: Prime Factorization
First, we need to find the prime factorization of 192. To do this, we divide the number by its smallest prime factors until we're left with only prime numbers.
192 ÷ 2 = 96
96 ÷ 2 = 48
48 ÷ 2 = 24
24 ÷ 2 = 12
12 ÷ 2 = 6
6 ÷ 2 = 3
The prime factorization of 192 is: 2 × 2 × 2 × 2 × 2 × 3 = 2^5 * 3.
Step 2: Grouping Factors
Next, we group the prime factors in pairs of two.
(2 × 2) × (2 × 2) × 2 × 3
Step 3: Simplifying the Radical
Now, we can take out the pairs of perfect squares from under the radical.
√[(2 × 2) × (2 × 2) × 2 × 3]
= √[(2 × 2)^2 × 2 × 3]
= (2 × 2)√(2 × 3)
= 4√6
Therefore, √192 is equal to 4√6 after simplifying the radical.