Prime Factorizations and Perfect Squares
1. Is the number a perfect square? Find each prime factorization to decide.
a) 468
b) 648
c) 2352
d) 3136
e) 99 225
Is it a perfect square?
a)
b)
c)
d)
e)
2. Evaluate each square root by finding the prime factorization first.
a) √225
b) √1225
c) √1764
d) √1089
3. Investigation: can you tell the ones digit of a perfect square from knowing the
ones digit of its square root?
a) Calculate the square of all these numbers with ones digit 3.
i) 32 =
ii) 132 =
iii) 232 =
iv) 332 =
v) 432 =
vi) 532 =
vii) 632 =
vii) 732 =
b) Make a conjecture: the square of a number with ones digit 3 has ones digit
___________.
c) Check your conjecture for these examples using a calculator
i) 24732 =
ii) 539832 =
iii) 709232 =
iv) 6204332=
v) your own example: _________ 2 = _____________
d) Complete the chart by squaring some numbers that end in each digit
Last digit of a number
Last digit of a square
0
1
2
3
4
5
6
7
8
9
e) Explain how you can know immediately that 765 095 643 772 843 is not a
perfect square.
4. Calculate the squares of 0, 5, 10, 15, and 20. Find the gaps in the pattern.
Then find the gaps in the gaps. Extend the pattern to determine the squares of
25, 30, 35 and 40.
Gaps in gaps:
Gaps:
Squares:
02
52
102
152
20 2
252
302
352
5. Use the Investigation (#3) to calculate square roots of perfect squares.
a) A perfect square ends in 4.
Its square root must end in ______ or ______.
b) A perfect square ends in 1.
Its square root must end in ______ or ______.
c) A perfect square ends in 9.
Its square root must end in ______ or ______.
d) A perfect square ends in 6.
Its square root must end in ______ or ______.
e) A perfect square ends in 5.
Its square root must end in ______.
f) A perfect square ends in 0.
Its square root must end in ______.
402
g) If you memorize the squares of 0, 5, 10, 15, 20, 25, 30, 35, and 40 (or else
use your answers to #4 to write down the pattern) then you can use the
above results to calculate the square root of any perfect square up to 1600.
Here is the pattern of squares:
02 = 0
52 = 25
102 = 100
152 = 225
202 = 400
252 = 625
302 = 900
352 = 1225
402 = 1600
Find the square root of 1024:
~ Since the ones digit is _______, what can the ones digit of the square root be?
_____ or ______.
~ Which multiples of 5 is the square root between? _______ and _______.
~ How do you know? ______________________________________________.
~ If the square root is a whole number, what must it be? ___________
~ Square your answer on paper (show your work). Do you get 1024?
6. Use the above method to find the square root of these perfect squares, and verify
your answers by multiplying and/or using a calculator, showing all your work:
i) 256
a) Number is between ______ and _______. (Squares of multiples of 5)
b) Square root must be between ________ and ________. (Multiples of 5)
c)Square root must end in _______ or ________, so it must be __________.
ii) 676
iii) 961
iv) 529
v) 784
7. Decide whether each number is a perfect square by determining what its square
root must be if it is a perfect square. Check your answers.
i) 296
ii) 541
iii) 676
v) 284
Since the square root of 25 is 5, the square root of 2500 must be 50.
Since the square root of 49 is _____, the square root of 4900 must be _____.
8. a) write two consecutive multiples of 10 that each square root lies between:
i) √5300
ii) √1231
iii) √6793
iv) √9999
v) √532
b) Find √2304 as follows:
Step 1: Find the two consecutive multiples of 10 that it lies between:
40 = √1600 < √2304 < √2500 = 50
Step 2: Find the possible ones digits of the square root:
√2304 ends in ______ or ______.
Step 3: Estimate – is the square most likely closer to 40 or 50? Predict √2304 and
then check your answer by multiplying.
c) Use the method above to estimate the square root of each perfect square below.
Then check your answer by multiplication.
i) √529
ii) √2401
iii) √441
iv) √1225
v) √2209
9. a) Explain why a perfect square that has ones digit 0 must also have tens digit 0.
b) Explain how you can know immediately that 54 376 598 720 is not a perfect
square.