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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.

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  1. anonymous
    • one year ago
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    @SolomonZelman can you help me again? ^ ^;

  2. mathstudent55
    • one year ago
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    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
    • one year ago
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    Alright.

  4. mathstudent55
    • one year ago
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    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
    • one year ago
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    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
    • one year ago
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    Are you familiar with any method of finding the prime factors of a number?

  7. anonymous
    • one year ago
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    @mathstudent55 can you help me with a quick question please :)

  8. anonymous
    • one year ago
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    I'm not fully aware No @mathstudent55

  9. mathstudent55
    • one year ago
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    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
    • one year ago
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    Here are the first several prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...

  11. mathstudent55
    • one year ago
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    Other than 2, all other even numbers are composite.

  12. anonymous
    • one year ago
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    Alright

  13. mathstudent55
    • one year ago
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    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
    • one year ago
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    Okay. Still not sure how to get 112.

  15. mathstudent55
    • one year ago
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    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
    • one year ago
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    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
    • one year ago
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    Alright thanks @mathstudent55

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