Take a factor out of the square root: 4 450
guessing at what you want .....
4√450
= 4√(25*9*2)
= 4√25√9√2
= 4*5*3√2= 60√2
Why did the square root go to the circus? To factor out its clown friend, 4!
To take a factor out of the square root, we factorize the number under the square root.
Let's factorize 450:
Start by finding the prime factors of 450:
450 ÷ 2 = 225
225 ÷ 3 = 75
75 ÷ 3 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1
The prime factors of 450 are 2, 3, 3, 5, and 5.
Now we can write 450 as a product of its prime factors:
450 = 2 × 3 × 3 × 5 × 5
Next, we identify pairs of identical prime factors:
450 = (2 × 3 × 3) × (5 × 5)
We can also rewrite this as:
450 = (2 × 3²) × (5²)
Since 3² = 9 and 5² = 25, we can rewrite the expression as:
450 = 2 × 9 × 25
Now, we can take the square root of the square factors outside the square root:
√(2 × 9 × 25) = √2 × √9 × √25
This simplifies to:
√(2 × 9 × 25) = √2 × 3 × 5
So, we can take the factor of 3 and 5 outside the square root:
√450 = 3 × 5√2
Therefore, we have taken out the factor of the square root of 450, and it can be written as 3 × 5√2.
To take out a factor from the square root, you need to determine if any perfect squares can be extracted from the number inside the square root symbol.
Let's start with the number 4. Since 4 is a perfect square (2 * 2 = 4), you can write it as the square of 2. Therefore, the square root of 4 can be expressed as √4 = 2.
When it comes to the number 450, we need to find its prime factorization and identify any perfect squares.
To find the prime factorization of 450, begin by dividing it by 2 until it is no longer divisible by 2:
450 ÷ 2 = 225
Then, keep dividing it by primes until you reach 1:
225 ÷ 3 = 75
75 ÷ 3 = 25
25 ÷ 5 = 5
So the prime factorization of 450 is 2 * 3 * 3 * 5 * 5.
Now, look for pairs of the same prime factors within the prime factorization. In this case, we have two 3's and two 5's. We can pair them up and take them out of the square root:
√(2 * 3 * 3 * 5 * 5) = √(2 * 3^2 * 5^2)
After taking out the pairs, you are left with:
√(2 * 3^2 * 5^2) = 3 * 5 * √2 = 15√2
Therefore, you can take out a factor of √2 from the square root of 450, resulting in 15√2.