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Finding Roots

Finding Roots. Composite Numbers. STEP 1:. Find all the factors of a number. Now let’s look at the numbers that are left. 1) 8. 8) 8. 2) 8. 3) 8. 4) 8. 5) 8. 6) 8. 7) 8. What are factors and how do I find them?. Example : Say you want to find the factors of 8.

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Finding Roots

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  1. Finding Roots Composite Numbers

  2. STEP 1: Find all the factors of a number

  3. Now let’s look at the numbers that are left 1) 8 8) 8 2) 8 3) 8 4) 8 5) 8 6) 8 7) 8 What are factors and how do I find them? Example: Say you want to find the factors of 8. The factors of 8are all the numbers that will divide into 8 evenly. (In other words, they are not decimals like 2.38 or 4.1) So take the numbers from 1 to 8 and divide 8 by each of them. 4 2 8 decimal If you get an answer with a decimal in it, the number you divided by is not a factor of 8, so cross out these answers. 1 decimal decimal decimal

  4. 8 4 1 2 1) 8 2) 8 4) 8 8) 8 The numbers on top are the factors of 8, factors:8, 4, 2 and 1 But, did you notice that the numbers on the top and the numbers you divided by (on the left) are the same? That’s because we are finding the factors two at a time, The number on the left and the number on top are both factors of 8. So to save time we don’t have to divide by every number from 1 to 8, we can go halfway and stop.

  5. If you write the factors of the number using the following system, you can see where your stopping point will be. All the factors of 8 are right here in this little box. 3) This is where they start to repeat, STOP HERE! You don’t need to write these repeating numbers down If we only have to find half of the factors, how do we know when we have gotten halfway and can stop? 1) Write the number with two little branches below it 8 2) Starting with ‘1 x 8’ Write all the pairs of factors that divide evenly into 8 1 * 8 2 * 4 4 * 2 8 * 1

  6. 12 48 32 81 We can stop checking numbers as soon as we reach this number Down the left side Up the right side You can stop here since there are no more numbers between these two factors on the bottom Here’s where the numbers start to repeat 4 * 3, etc. so stop here. Stop, since the next number is 8 Make sure you check all the numbers up to the number on the bottom right, this is where they start to repeat. Practice: Find the factors of the following numbers Read the factors in this order 1 * 48 2 * 24 3 * 16 4 * 12 6 * 8 7 * decimal 1 * 12 2 * 6 3 * 4 1 * 32 2 * 16 4 * 8 1 * 81 3 * 27 9 * 9 5 * decimal 6 * decimal 7 * decimal 8 * 4(repeat) Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 81: 1, 3, 9, 27, 81 Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 32: 1, 2, 4, 8, 16, 32

  7. Step 2: Split the number into two factors This is a square root, so look for perfect squares Use the splitting property to simplify the following: 1) Find all the pairs of factors-look for perfect squares 2) Find the pair with the largest perfect square 1 * 12 2 * 6 3 * 4 3) Write this pair in the following order: 4 is a perfect square Answer 4) Take the square root of the perfect number

  8. Step 2: Split the number into two factors This is a square root, so look for perfect squares Use the splitting property to simplify the following: 1) Find all the pairs of factors-look for perfect squares 2) Find the pair with the largest perfect square 1 * 32 2 * 16 4 * 8 3) Write this pair in the following order: 4 and 16 are perfect squares Answer 4) Take the square root of the perfect number

  9. Step 2: Split the number into two factors This is a square root, so look for perfect squares Use the splitting property to simplify the following: 1) Find all the pairs of factors-look for perfect squares 1 * 81 3 * 27 9 * 9 2) Double factors like this mean that the original numberwas a perfect square and this splitting process is unnecessary. 3) Take the square root of 81 (see perfect numbers chart) Answer Note: Checking for Prime numbers should also be done before trying the splitting process because prime numbers cannot be broken up at all. Example of prime root: Answer

  10. Step 2: Split the number into two factors This is a square root, so look for perfect squares Use the splitting property to simplify the following: 1) Find all the pairs of factors-look for perfect squares 2) Find the pair with the largest perfect square 3) Write this pair in the following order: 1 * 48 2 * 24 3 * 16 4 * 12 6 * 8 4 and 16 are perfect squares Answer 4) Take the square root of the perfect number

  11. Step 2: Split the number into two factors This is a cube root, so look for perfect cubes Use the splitting property to simplify the following: 1) Find all the pairs of factors-look for perfect cubes 2) Find the pair with the largest perfect cube 1 * 108 2 * 54 3 * 36 4 * 27 6 * 18 9 * 12 3) Write this pair in the following order: 27 is a perfect cube Answer 4) Take the cube root of the perfect number

  12. Practice Problems

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