1.6 Square Roots • a calculator

1.6 Square Roots

You will need

? grid paper ? a calculator

GOAL

Estimate and calculate the square root of a whole number.

Learn about the Math

Sandra and her father cut a hole in the ice on a lake to measure the thickness of the ice. Then they used a formula with a square root (1) to determine if 30 cm of ice could safely support their total mass of 125 kg.

Required thickness 0.38 load in kilograms

square root (0)

one of two equal factors of a number; for example, the square root of 100 is represented as 1 00 and is equal to 10 because 10 10 or 102 100

Communication Tip

? Is the ice thick enough to support the total mass of Sandra and her father?

? In many formulas, the multiplication symbol is not included. For example, 0.38 represents 0.38 .

? You can use the symbol to show that a number is approximately equal to another number. For example, 2 1.414.

A. Draw a 10-by-10 square, an 11-by-11 square, and a 12-by-12 square on grid paper. Calculate the area of each square.

B. How can you calculate the length of each side of a square if you know only the area of the square?

C. Does a square with an area of 125 square units have whole-number dimensions? Explain your reasoning.

D. How can you use the side lengths of the squares you drew in step A to estimate 1 25?

E. Use a calculator to determine 1 25. Round your answer to two decimal places.

28 Chapter 1

Calculator Tip

Different calculators require different key sequences to calculate square roots. TI-15:

125 ? Other calculators: 125

NEL

F. Is the ice thick enough to support the total mass of Sandra and her father? Show your work.

Reflecting

1. Explain how to use the square key ? or the power key F on your calculator to check your answer in step E.

2. When you square your answer in step E, why do you not get exactly 125?

Work with the Math

Example 1: Estimating a square root by squaring numbers

The area of a square floor is 82.0 m2. Estimate the length of each side of the floor.

Reilly's Solution

82.0 m2

m

m

82.0 2 82.0

92 81

9.12 82

9.1 82

I can find the side length of a square with an area of 82.0 square units by calculating 82.

The square root of 81 is 9. So, the square root of 82 must be a bit more than 9. I picked 9.1.

The side length of the square is between 9.0 and 9.1 m2.

Example 2: Using the square root key on a calculator

The mass of a truck is 5000 kg. What thickness of ice is needed to support the truck?

Tamara's Solution

First I estimated the thickness: 0.38 5000. 5 000 must be close to 70 because 702 4900. Multiplying 70 by 0.38 is less than half of 70, or about 30 cm.

Then I used a calculator to calculate the thickness.

Using the TI-15: .38 C 5000 ?

Using another calculator:

5000

C .38 G

The ice needs to be about 27 cm thick to support the truck.

NEL

Number Relationships 29

A Checking

3. Estimate each square root to one decimal

place.

a) 15

c) 50

b) 3 00

d) 1 22

4. State whether the square root is a whole

number. Then calculate the square root.

Round to three decimal places, if necessary.

a) 42

c) 9 61

b) 1 44

d) 2 052

5. Use the following formula to estimate the thickness of ice, in centimetres, needed to support each vehicle. Thickness (cm) 0.38 load (kg) a) a car with a mass of 800 kg

b) a truck with a mass of 1800 kg

B Practising

6. Use estimation to determine whether or not each answer is reasonable. Use the square root key on your calculator to correct any unreasonable answers. a) 8 2.8 d) 342 28.5 b) 10 3.2 e) 1482 38.5 c) 289 27 f) 3052 55.2

7. Calculate. a) 18 b) 75

c) 1 50 d) 38

e) 8 00 f) 3 900

8. A square field has an area of 3000 m2.

3000 m2

a) Is the side length of the field a whole number of metres? Explain how you know.

b) Estimate the side length of the square.

c) How do you know that the side length is between 50 m and 60 m?

d) Calculate the side length of the square field.

9. Explain how you know that 71 is between 8 and 9.

10. Explain how you can use squaring to estimate 7.

11. Use mental math to calculate each square root. Then use squaring to check your answer. a) 1 00 c) 4 00 e) 1 600 b) 1 44 d) 9 00 f) 3 600

12. Use the following formula to estimate the time an object takes to fall from each height below.

Time (s) 0.45h eight ( m)

a) 100 m

d) 900 m

b) 200 m

e) 2000 m

c) 400 m

f) 10 000 m

13. Nico squared some numbers and got these answers. Use mental math to determine each number she squared. Explain your reasoning.

a) 49

c) 30

e) 169

b) 1225

d) 72

f) 625

30 Chapter 1

NEL

14. a) Choose a number.

b) Name a number that has a square root less than your chosen number.

c) Name the number whose square root is your chosen number.

d) Name a number that has a square root greater than your chosen number.

15. a) Try Tamara's number trick: ? Choose any whole number greater than 0. ? Square it. ? Add twice your original number. ? Add 1. ? Calculate the square root of the sum. ? Subtract your original number. ? Record your answer.

b) Try Tamara's number trick with four other numbers.

c) What do you notice about your answers in parts (a) and (b)?

16. The year 1936 is the last year whose square root was a whole number. What is the next year whose square root will be a whole number? Explain your reasoning.

17. Examine each number. What number can you add to this number to make a number that has a whole-number square root?

a) 42

c) 470

b) 101

d) 1000

18. A palindrome is a number that is the same when it is read from left to right and from right to left. Both 14 641 and its square root, 121, are palindromes. Find at least three other numbers that are palindromes and have square roots that are palindromes.

19. a) Calculate these square roots.

5

5 0 000

500

5 000 000

b) Describe any patterns you see in part (a).

c) Identify the next calculation in the pattern.

C Extending

20. Ken calculated the square root of a number. Then he calculated the square root of the square root. His answer was 25. What was his original number? Explain your reasoning.

21. a) Choose three different two-digit numbers. Determine the prime factorization of each number.

b) Calculate the square of each number in part (a). Determine the prime factorization of each square.

c) Compare the prime factorization of each square with the prime factorization of its square root. How can you use the prime factorization of a square to calculate its square root?

d) Use the prime factorization of 23 409 to calculate its square root. 23 409 34 172

22. a) Try Sheree's number trick: ? Choose any two-digit number. ? Subtract 2 from this number. ? Calculate the product of the two numbers. ? Add 1 to the product. ? Calculate the square root of the sum.

b) Repeat Sheree's number trick with another two-digit number.

c) What do you notice about your answers in parts (a) and (b)?

NEL

Number Relationships 31

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