This picture is a way you can use prime factorization to determine the square root of 225.
What I did in the picture is that I made a factor tree. First I started off with the number that was given in the question that we had to find the square root. Then you find 2 factors that equal to 225. I chose 15x15. but you can use different ones if you'd like because you will still end up with the same prime factors! Next you have to find out whether 15 is a prime number. Well it's not, 3 and 5 are factors of 15. Next you figure out whether 3 and 5 are prime numbers. And they are! Now you just have to make the upside down hut and separate the prime factors equally. To prove you work you can make a diagram. Like what I did above showing how it is a perfect square.
8) Determine the prime factorization of each number. Which numbers are perfect squares?
a) 144 b)60 c)40
A)In this picture I did pretty much the same thing as I did in the first question I explained about above.
I started off at 144 then found out that 12 and 12 are the factors of 144. 12 is not a prime number so I found the factors of 12. I chose 6 and 2. Next I found that 6 isn't a prime number so you have to find the factors of 6 which is 3 and 2. Now we find out that 3 and 2 are not prime numbers so your tree stops there.
Then you separate them equally once again. This is a perfect square because it has a square root, and that you can have the same amount when you separate the number equally and multiply them. So 3x2x2=12 and 3x2x2=12. Then 12x12 = 144 or 12². All sides must have the same length in order for it to be a square. Which it is.
B)I found the 2 factors of 60 which were 10 and 6. Then I knew that 10 and 6 are not prime numbers. So for 10 the factors are 2 and 5 and for 6 it is 3 and 2. Now you end up with prime factors at the bottom and your tree is done.
C) I found the factors of 40 and I chose 4 and 10. I knew that 10 and 4 are not prime numbers. So for 10 the factors are 2 and 5. For 4 2 and 2 are the factors. 2 and 5 are prime numbers so that is the end of your factor tree.
15)A. Evaluate the square root of 49
The square root is seven because a number that is multiplied by itself and equals that number is the square root. And if you did a factor tree the factors would be 7 and 7. 7 is a prime number so you your factor tree is done. To prove it you put 7 for one side of a square and the other factor which is seven also on the other side now think if 7 is the same as 7. It is right? So now 7 multiplied by 7 is 49 or 7².
16)A. Find the square root of 9
The square root of 9 is 3 because in a factor tree the factors of 9 are 3. Since 3 is a prime number the factor tree is done. To prove this I made a square diagram that show it has equal sides with the "tickies" so the sides must be the same in order for is to be a square and to be a square root of a number. So you divide the numbers evenly and you get 3 on one side and 3 on the other. Now think, is 3 the same as 3? Of coarse it is! So 3x3 or 3² gets you nine.
17) A fridge magnet has an area of 54mm ². Is 54 a perfect square? Use prime factorization to find the answer.
54 is not a perfect square because the factors of 54 is 9 and 6. Since they are not prime numbers you have to continue. So 9's factors are 3 amd 6's factors are 2 and 3. 2 and 3 are prime numbers so thats the end. Now to find out whether it is a perfect square you have to separate the number equally so 3x2 and 3x3. Then you find out 3x2 and 3x3 is. The answers are 6 and 9. 6 is not the same as 9 so it can not be a square it is a rectangle!. So it can not be a perfect square.
20)Adam's uncle has instructions for building a shed. One page of instructions is not very clear.
A) What is the area of a rectangle?
B) What is the length of the side of the square?
A) In the picture the length of the rectangle was 9m and the width was 4m. So the formula to find the area is
l x w. So now you get the length which is 9 and the width which was 4 and multiply them. 9x4 = 36m².
B) Because the area of the rectangle was the 36m² The area for the square has to be the same thing. So instead of 9 x 4 what other factor can you use to just square and equal to 36. 6 is the answer. Here is a factor tree that I will show you, that will prove that 6 is the length of one side.
Here is a video about square roots :)
1) Pythagorean Relationship
- The Pythagorean Relationship is the relationships between the lengths of the sides of a right triangle.
- The sum of the areas of the squares attached to the legs of a right triangle equals the area attached to the hypotenuse.
~ A right triangle has a right angle of 90 degrees, and this may be marked as a square.
2. Solve for the missing side length.
3. Is this a right triangle? Prove it!!!