The square root of 4356 is an integer. Without a calculator, determine what the integer is by eliminating possibilities. Do the same for 8464.

Question

I do not get how to solve this problem

Answer 1

4356 is a number that lies between two perfect square integers: 3600 and 4900. 

 

√(3600) = 60

 

√(4900) = 70

 

 

So the square root of 4356 must be between 60 and 70. 

 

 

Let's take the median of these two squares and two square-roots to see how far 4356 is in between the integers.

 

(3600 + 4900) / 2 = 8500 / 2 = 4250

 

(60 + 70) / 2 = 130 / 2 = 65

 

 

The median of the squares is 4250.  This is close to the actual number 4356.

The median of the integer's square-roots is 65. 

 

4356 is between 4250 and 4900.  Now this narrows it down.  The square-root of 4356 is strictly between 65 and 70.

 

 

Since the units digit of 4356 is the product of 6² after carrying the 3 over to the tens place in multiplication, the only possible number between 65 and 70 is 66.

 

 

   66

× 66

____

   

 

The square-root of 4356 is 66.

 

 

Answer 2

Emma,
 
You write "I don't get it" far too often.  PLZ try, "With some help, I want to get it" followed by "Oh, now I got it!"
 
Square roots are important in math.  Most math students memorize the squares (and, therefore, the square roots) of numbers from 0 to 20 (or at least 0 to 10). Study these:
 
02 = 0
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
 
 
So, what is the square root of 4356.  First, 4356 is four digits, so we know that the square root is 2-digits long because the biggest 1-digit number squared is 81 and the smallest 3-digit number (100) squared is 10000.
 
Now, think about how you multiply two numbers (that is, a number by itself) -- you put down a digit and carry another digit to add to the next position. Look at the list above and find which ending digits, when squared, produce a final 6.  Looks like we know that the last digit of the number we want is either a 4 or a 6.
 
Now, PLZ remember more about long multiplication (and, when we are done, you might want to work this problem backward by squaring the answer).  The 10's digit is multiplied by the 10's digit and added to the carry to form the first 2 digits of the product.  He we are wanting to find some square that is smaller than 43.
 
The problem said to "eliminate possibilities," so let's pick 6 (whose square of 36 is less than 43) and we already have either 4 or 6 as the units digit.
    642 = 4096    (TMI:  I'm a retired computer pro, and any computer pro knows this)
    662 = 4356    Ah-ha!!
 
That's not too bad for just a few possibilities.  The key is to be smart about eliminating possibilities.
 
Now, for 8464, what squares end with a "4" ??

The square root must be between 90² = 8100 and 100² = 10000.
And, what squares are a little lower than 84 (because we must add the carry, remember) ?
You should have only a couple of possibilities.
 
Go for it !