√176, √44 , √352, √88
√176 ≈ 13.2664991614216
√44 ≈ 6.6332495807108
√352 ≈ 18.7616632871837
√88 ≈ 9.38083164359185
To simplify these square roots, we can look for the largest perfect square that divides each number inside the square root.
1. √176:
The largest perfect square that divides 176 is 16 since 16 * 11 = 176. We can write √176 as √(16 * 11).
Simplifying further, we can take the square root of 16, which is 4, and move it outside the square root symbol. So, √(16 * 11) becomes 4√11.
Therefore, √176 simplifies to 4√11.
2. √44:
The largest perfect square that divides 44 is 4 since 4 * 11 = 44. We can write √44 as √(4 * 11).
Simplifying further, we can take the square root of 4, which is 2, and move it outside the square root symbol. So, √(4 * 11) becomes 2√11.
Therefore, √44 simplifies to 2√11.
3. √352:
The largest perfect square that divides 352 is 16 since 16 * 22 = 352. We can write √352 as √(16 * 22).
Simplifying further, we can take the square root of 16, which is 4, and move it outside the square root symbol. So, √(16 * 22) becomes 4√22.
Therefore, √352 simplifies to 4√22.
4. √88:
The largest perfect square that divides 88 is 4 since 4 * 22 = 88. We can write √88 as √(4 * 22).
Simplifying further, we can take the square root of 4, which is 2, and move it outside the square root symbol. So, √(4 * 22) becomes 2√22.
Therefore, √88 simplifies to 2√22.
To summarize, the simplified forms of the given square roots are:
√176 = 4√11
√44 = 2√11
√352 = 4√22
√88 = 2√22
To find the square root of a number, there are a few different methods you can use. One common method is to use the prime factorization of the number.
Let's start with √176.
Step 1: Prime Factorization of 176
To find the prime factorization of 176, we start by dividing it by the smallest prime number, which is 2.
176 ÷ 2 = 88
Now, we continue dividing 88 by 2 until we can't divide anymore.
88 ÷ 2 = 44
44 ÷ 2 = 22
Now, we move to the next prime number, which is 3. But 22 is not divisible by 3.
Next, we move to the next prime number, which is 5. Again, 22 is not divisible by 5.
Now, we move to the next prime number, which is 7. And 22 is divisible by 7.
22 ÷ 7 = 3
So, the prime factorization of 176 is 2^4 × 7 × 1.
Step 2: Simplify the Square Root
Now that we have the prime factorization of 176, we can simplify the square root.
√176 = √(2^4 × 7 × 1)
Using the property of square root that says √(ab) = √a × √b, we can separate the square root into individual factors.
√176 = √(2^4) × √7 × √1
Simplifying further:
√176 = 2^2 × √7
= 4√7
So, the square root of 176 is 4√7.
Now, let's move on to the other numbers.
√44:
Following the same steps as above, we find that the prime factorization of 44 is 2^2 × 11.
Therefore, √44 = √(2^2 × 11) = 2√11.
√352:
The prime factorization of 352 is 2^5 × 11.
Therefore, √352 = √(2^5 × 11) = 4√22.
√88:
The prime factorization of 88 is 2^3 × 11.
Therefore, √88 = √(2^3 × 11) = 2√22.
So the simplified square roots are:
√176 = 4√7
√44 = 2√11
√352 = 4√22
√88 = 2√22