ordering radicals from smallest to biggest √176, √44 , √352 , √88
√44, √88, √176, √352
To order these radicals from smallest to biggest, we need to simplify them first.
Let's start by simplifying each of the radicals:
√176 = √(16 × 11) = √(16) × √(11) = 4√11
√44 = √(4 × 11) = √(4) × √(11) = 2√11
√352 = √(16 × 22) = √(16) × √(22) = 4√22
√88 = √(4 × 22) = √(4) × √(22) = 2√22
Now, we can clearly see that the simplified radicals are:
2√11, 4√11, 2√22, 4√22
To order them from smallest to biggest, we look at the coefficients first:
2√11, 2√22, 4√11, 4√22
Next, we compare the numbers inside the radicals:
2√11, 2√22, 4√11, 4√22
Since 11 is smaller than 22, we can say that 2√11 is smaller than 2√22.
Finally, we compare the coefficients again:
2√11, 2√22, 4√11, 4√22
The final order from smallest to biggest is:
2√11, 2√22, 4√11, 4√22
To order the given radicals from smallest to biggest (√176, √44, √352, √88), we need to simplify or estimate the value of each radical.
Here's how you can do it step by step:
1. Simplify each radical by finding perfect square factors:
- √176: Rewrite 176 as a product of its prime factors: 2 × 2 × 2 × 2 × 11. Now, group the prime factors in pairs of perfect square factors: 2 × 2 × 4 × 11. Taking the square roots of the perfect square factors, we get 2 × 2 × √11 = 4√11.
- √44: This number is already simplified since 44 = 2 × 2 × 11. So, √44 = 2√11.
- √352: Again, rewrite 352 as a product of prime factors: 2 × 2 × 2 × 2 × 22. Group the factors into pairs: 2 × 2 × 4 × 22. Taking the square roots, we get 2 × 2 × √22 = 4√22.
- √88: Rewrite 88 as 2 × 2 × 2 × 11. Group into pairs: 2 × 2 × 8 × 11. Taking the square roots, we have 2 × 2 × √8 × √11 = 4√(2 × 2 × 2) × √11 = 8√11.
2. Now that we have simplified forms of each radical:
- √176 ≈ 4√11
- √44 = 2√11
- √352 ≈ 4√22
- √88 = 8√11
3. Compare the simplified radicals and order them from smallest to largest:
- √44 < √176 < √352 < √88
- 2√11 < 4√11 < 4√22 < 8√11
So, the ordered list is: √44, √176, √352, √88.