## anonymous one year ago Can you show me how to do this? A square picture frame occupies an area of 112 ft2. What is the length of each side of the picture in simplified radical form? Show your work.

1. anonymous

@SolomonZelman can you help me again? ^ ^;

2. mathstudent55

Since the picture frame is a square, and the area is 112 ft^2, then each side is $$\sqrt {112~ft^2} = \sqrt{112} ~ft$$. Now we need to simplify $$\sqrt{112}$$.

3. anonymous

Alright.

4. mathstudent55

In order to simplify a root, you need to find the largest perfect square factor of the number whose root you are taking. What is the largest perfect square factor of 112?

5. mathstudent55

For example, if we were simplifying $$\sqrt{18}$$, you could easily see that the largest perfect square factor of 18 is 9. 18 is an easy number to work with. 112 is not so easy. In order to figure it out, we find all the prime factors of 112.

6. mathstudent55

Are you familiar with any method of finding the prime factors of a number?

7. anonymous

@mathstudent55 can you help me with a quick question please :)

8. anonymous

I'm not fully aware No @mathstudent55

9. mathstudent55

Ok. I'll show you. Every whole number, 2 or larger is either a prime number or a composite number. A prime number is a number that can be divided evenly by only 2 different numbers, 1 and the number itself. A composite number can be divided evenly by more than 2 numbers. Every composite number is a product of prime numbers.

10. mathstudent55

Here are the first several prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...

11. mathstudent55

Other than 2, all other even numbers are composite.

12. anonymous

Alright

13. mathstudent55

For example, the number 17 is prime. It is only divisible by 1 and 17. 18 is composite. It is divisible by 1, 2, 3, 6, 9, 18. In addition, since 18 is composite, it can be written as a product pf prime numbers. 18 = 2 * 3 * 3

14. anonymous

Okay. Still not sure how to get 112.

15. mathstudent55

Now we need to see which prime numbers have a product of 112. To find the prime factorization of 112, we start with 112 and divide it by 2, the smalles prime number. 112/2 = 56 We try to keep dividing by 2 as many times as is possible. 56/2 = 28 28/2 = 14 14/2 = 7 Since 7 is no longer divisible by 2, we move on to the next higher prime number that 7 is divisible by. Since 7 is prime, then 7 can be divided by 7. 7/7 = 1 You must always finish with 1. Now we see all the divisions we did to 1112 to end up with 1, and we have the prime factors of 112. 112 = 2 * 2 * 2 * 2 * 7

16. mathstudent55

Notice that 2 * 2 * 2 * 2 = 16, so 112 = 16 * 7, and 16 is a perfect square. $$\sqrt{112} = \sqrt{16 \times 7} = \sqrt{16}\times \sqrt{7} = 4 \sqrt {7}$$

17. anonymous

Alright thanks @mathstudent55