anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
@SolomonZelman can you help me again? ^ ^;
mathstudent55
  • 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 \(\sqrt{112} \).
anonymous
  • anonymous
Alright.

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mathstudent55
  • 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?
mathstudent55
  • 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.
mathstudent55
  • mathstudent55
Are you familiar with any method of finding the prime factors of a number?
anonymous
  • anonymous
@mathstudent55 can you help me with a quick question please :)
anonymous
  • anonymous
I'm not fully aware No @mathstudent55
mathstudent55
  • 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.
mathstudent55
  • mathstudent55
Here are the first several prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...
mathstudent55
  • mathstudent55
Other than 2, all other even numbers are composite.
anonymous
  • anonymous
Alright
mathstudent55
  • 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
anonymous
  • anonymous
Okay. Still not sure how to get 112.
mathstudent55
  • 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
mathstudent55
  • mathstudent55
Notice that 2 * 2 * 2 * 2 = 16, so 112 = 16 * 7, and 16 is a perfect square. \(\sqrt{112} = \sqrt{16 \times 7} = \sqrt{16}\times \sqrt{7} = 4 \sqrt {7}\)
anonymous
  • anonymous
Alright thanks @mathstudent55

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