MPM2D
Ms. Kueh
Do not use a calculator for this sheet!
A _______________________________ is the result of multiplying a whole number by itself.
For example 64 is a perfect square because it is ______________
List three perfect squares between 1000 and 2000.
The ______________________________ of a number is what you multiply by itself to get the number.
For example, 8 is the square root of 64 because _____________________
*Squaring and Square rooting are reverse operations!
Example 1 What picture would you draw to show why √16 is 4 ?
Example 2 What does 5 3 mean? What picture could you draw?
Example 3 Why does it make sense that √250 000 is 100 times as much as √25 ?
We can use facts like 10 × 10 = and 100 × 100 = to help estimate the square roots of larger numbers.
For example, since √15 is close to then √1500 is close to and √150 000 is close to

Example 5 Estimate each square root without a calculator. a) √88 b) √110 c) √3600 d) √4900 e) √30 f) √3000
Example 6 Explain how you know that 250 cannot be a perfect square.
Example 7 Which of these powers of 10 are perfect squares? Explain.
100 1000 10 000 100 000
Example 8 What is the least number you could multiply 2 × 2 × 4 × 5 by to make a perfect square? Explain.
Example 9 You want to multiply 2 × 2 × 5 × 3 by a number to make a perfect square. List three possible amounts you could multiply by and prove that each is a perfect square.
Example 10 How could factoring 57 600 as 64 × 36 × 25 help you figure out its square root?

*Powers of whole numbers grow very quickly. For example,
3 4 = 81
3 5
3 6
= 243
= 729
This is why there is the saying that something “grows exponentially” means that it increases faster and faster and faster
*Powers of fractions or decimals less than 1 shrink as the exponent increases. For example,
(
1
2
)
2
=
(
1
2
)
3
=
(
1
2
)
4
=
*Powers of negative numbers
(−2) 2
(−2) 3
(−2) 4
(−2) 5
=
=
=
=
What conclusion can you make?
Example 12 a) Order from least to greatest:
(
1
3
)
3
(
1
3
)
4 b) Order from least to greatest:
2 3 2 4
(
2
1
3
2
)
2
(
2
1
3
5
)
5 c) What do you notice if you compare the answers to a) and b) ?

Example 13 a) What is √49 ? b) Solve 𝑥 2 = 49 c) Explain the difference between a) and b) d) What is √−36 ?
Number systems:
3 e) What is √−8 ? f) Why can you cube root a negative number when you can’t square root a negative number in the real number system? g) What 𝑛 -th roots can you calculate in the real number system if you have a negative number?