### 6 AM Quiz (11 June 2011)... There are more than one way!

What number multiplied by itself is equal to the product of 32 and 162?

There are 2 or more ways to solve this problem. Describe at least 2 methods :)

1. Method 1:
32*162=5 184
√5 184=72

Method 2:
guess and check:
32*162=5 184
1*1=1
2*2=4
3*3=6
4*4=16
5*5=25
6*6=36
7*7=49
8*8=64
..
..
..
72*72=5 184

2. Therefore, the answer is 72.

Ishani[04]

3. 1. Product of 32 and 162 = 5184
Square root of 5184 = 72
Check:72*72 = 5184

2. Prime factorisation of 32 = 2^5
Prime factorisation of 162 = 2*3^4
Square root of 2^6*3^4 = Square root of 72*72
= 72

4. Method 1: 32x162= 5148
Square root 5148= 72

Method 2: Prime factorisation of 32= 2^2^2^2^2
Prime factorisation of 162= 2^3^3^3^3
2^6= 64
3^4= 81
81-64= 17
Median of 1~17= 9
81-9=72 (Not useful as 64+9= 73, which the mean of 72 and 73 would be 72.5)

Chan Shawn Kit(10)

5. 32*162=5148

1. Square root of 5148 = 72
72*72=5148

2. Prime factorisation of 32 and 162.
32 = 2*2*2*2*2
= 2^5
162 = 2*3*3*3*3
= 2*3^4
2^5*2*3^4=2^6*3^4
When 2^6*3^4 is square rooted, it will be 2^3*3^2 which equals to 72.

6. Prime factorisation: 32=2*2*2*2*2
=2^5
162=2*3*3*3*3
=2*3^4
32*162=2^6*3^4
=(2^3*3^2)*(2^3*3^2)
2^3*3^2=8*9
=72

Square root: 32*162=5148
The square root of 5148 is 72 threfore, 72*72 is 5148.

7. 32*162=5148
The square root of 5148 is 72.

Ding Nina Lin

8. M E T H O D O N E :

32 x 162 = 5184
Squareroot 5184 and you'll arrive at 72.

M E T H O D T W O :

Prime Factorise 32 and 162:
32 = 2 x 2 x 2 x 2 x 2
= 2^5

162 = 2 x 3 x 3 x 3 x 3
= 2 x 3^4

Add both of the Prime Factorisations together:
(2^5) + (2 x 3^4)
= 2^6 x 3^4

Assume that (2^6 x 3^4) is ‘2 sets’. And the ‘2 sets’ would be the product of 32 and 162.
However, we only want ‘1 set’.

2 sets = 2^6 x 3^4
Thus, 1 set = 2^3 x 3^2
= 72

9. What number multiplied by itself is equal to the product of 32 and 162?

Method 1:
32*162=5184
√5184=72

Method 2:
32=2*2*2*2*2
162=2*3*3*3*3
32*162=2*2*2*2*2*2*3*3*3*3
Also can be presented in this way...
32*162=(2*2*2*3*3)*(2*2*2*3*3)
√32*162=2*2*2*3*3
√32*162=(2^3)*(3^2)
√32*162=72

Lovy

10. Method One :

32 x 162 = 5184

Squareroot 5184 and you will get 72.

Method Two :

Prime Factorise - 32 and 162

32 = 2 x 2 x 2 x 2 x 2
= 2^5
162 = 2 x 3 x 3 x 3 x 3
= 2 x 3^4
162 x 32 = 2^6 x 3^4
Squareroot (2^6 x 3^4)
= 2^3 x 3^2
= 8 x 9
72

11. Method One :

Find the product of 32 and 162 . 32*162=5184
Square root the product which is 5184 and you will get 72 .

Ans : 72

Method Two :

Prime Factorisation Method.

Prime Factorise 32 and 162.

32=2*2*2*2*2
= 2^5

162= 2*3*3*3*3
= 2*3^4

32*162=2^5 x 2 x 3^4

Square Root (32*162) = distribute the prime numbers out equally into

groups of two (2^5 x 2 x 3^4)
= (2*2*2*3*3) x (2*2*2*3*3)
= 72 x 72
Ans : 72 .

12. Method 1:
Square Root
32x162=5184
√5184=72
Ans:72
Method 2:
32
/ \
2 16
/ \
2 8
/ \
2 4
/ \
2 2
162
/ \
2 81
/ \
3 27
/ \
3 9
/ \
3 3
32=2^5
162=2*3^4
162*32=2^6*3^4
√2^6*3^4=(2^3*3^2)
2^3*3^2=72

13. Method 1

32x162=5184
√5184=72
A.n.s. 72

Method 2

32=2*2*2*2*2
162=2*3*3*3*3
√(2*2*2*2*2*2*3*3*3*3)=√[(2*2*2*3*3)*(2*2*2*3*3)*]
=2^3*3^2
=8*9
=72
A.n.s. 72

14. first method:
32x162=5184
√5184=72

second method:
prime factorization of:
32=2*2*2*2*2
162=2*3*3*3*3
32*162=2*2*2*2*2*2*3*3*3*3
square rooting=dividing the number into 2 groups of the same number
so, (2*2*2*3)*(2*2*2*3)=5184
2*2*2*3=√5184=72

15. Method 1
32*162=5184
√5184 = 72
ans: 72
Method 2
Prime factorisation
32=2*2*2*2*2
162=2*3*3*3*3
32*162=2*2*2*2*2*2*3*3*3*3
√32*162=(2*2*2*3*3) * (2*2*2*3*3)
= 72*72
ans: 72

16. method 1:
32*162 = 5184
square root of 5184 = 72

method 2:
[Using prime factorisation]
32 = 2*2*2*2*2
162 = 2*3*3*3*3
32*162 = 2*2*2*2*2 * 2*3*3*3*3
square root of 32*162 = (2*2*2*3*3) * (2*2*2*3*3)
= 72 * 72
Ans:72

17. -First Method:

Find the product of 32 and 162
=5184

Square Root the product
=72

-Second Method:

Prime Factorise both of the numbers
= 2^5, 2x3^4

Combine them together and square root it
=2^3x3^2
=72

18. method 1:
32x162 = 5184
square root of 5184 = 72

method 2:
[Using prime factorisation]
32 = 2x2x2x2x2
162 = 2x3x3x3x3
32*162 = 2x2x2x2x2 x 2x3x3x3x3
32x162 = (2x2x2x3x3) x (2x2x2x3x3)
= 72 x 72
Ans:72

19. 1.Square root

32x162=5184

√5184=72

72x72=5184

ANS:72

2.Prime factorisation

32-2*2*2*2*2

162-2*3*3*3*3

32x162=2*2*2*2*2*2*3*3*3*3

=(2*2*2*3*3)x(2*2*2*3*3)

=72x72
ANS:72

20. What number multiplied by itself is equal to the product of 32 and 162?
Method 1:
Multiply 32 and 162 together --> 32x162=5184
√5184 =72

Method 2:
Simplifying both numbers into prime numbers; 32 162
32=2x2x2x2x2 = 2^5

162= 2x3x3x3x3 = 2x 3^4

32x162= 2^5 x 2 x 3^4
= 2^6 x 3^4

By classifying them into groups of 2,
2^6 x 3^4 = (2^3 x 3^2) x (2^3 x 3^2)
(2^3 x 3^2) = 72

21. Owen here ._.

22. @Ishani: You are right that Guess and Check is another method we could adopt :)
It would be clearer if you indicated that it was SYSTEMATIC Guess and Check and further describe why you choose to organise the numbers in a certain pattern :)

@Shawn Kit: How does "Median" come into the picture? Maybe you could help us to make the connection?

@Toby: We can't "Add" both of the Prime Factorisations together like what you've described: (2^5) + (2 x 3^4)
Instead, it should be written as
32 x 162 - (2^5) x (2 x 3^4)

Indeed, many of you are able to describe the "How" to do the square root using the Prime Factorisation method. On the other hand, only Mavis, Ryan and Owen have explicitly explained how to do the square root, which is, organising the factors into 2 groups. Well Done to all 3 :)

23. 1) 32*162 = 5184
Square root of 5184 = 72
Ans: 72

2) Prime Factorisation
Prime factorise 32 and 162
32=2*2*2*2*2= 2^5
162=2*3*3*3*3= 2*3^4
32*162=2^6*3^4

Then we split them into equal groups.
2^6*3^4= (2^3*3^4) x (2^3*3^4)
2*group= 162
1*group= (2^3*3^4)= 72

Ans: 72

24. 1) 32*162 = 5184
Square root of 5184 = 72

2)32 = 2^5
162 = 2*3^4
2^5*2*3^4=2^6*3^4

To find the square root of this number, we must split the prime factors into 2 equal groups.
2^6*3^4=(2^3*3^2) x (2^3*3^2)
(2^3*3^2) = 72