1. Square Roots and Irrational Numbers
Grade 7 Pre-Algebra
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Let’s warm up :
1) Find the area of a trapezoid with Bases: cm and 16 cm
Height: 10 cm
2) Find the area of a triangle with
Base: cm
Height: 6 cm
3) Find the surface area of a cylinder with Radius: 6 cm
Height: 25 cm
4) Find the volume of a cylinder with
Radius : 6 cm
Height: 25 cm
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3. Square Roots and Irrational Numbers Definition of square root
Today we will discuss about
Square Roots and Irrational Numbers
Definition of square root
In words: The square root of a number is one of its two equal factors.
In Symbols: If x2 = y, then x is a square root of y.
For example: The square root of 49 is 7 since 7.7 or 72 is 49. It is also true that -7.(-7) = 49. So -7 is another square root of 49.
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4. The Square Root Symbol
The symbol √, called the radical sign is used to indicate a non-negative square root.
√49 indicates the non-negative square root of 49.
√49 = 7
-√49 indicates the negative square root of 49.
-√49 = -7
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Find each square root.
a.) √25
The symbol √25 represents the nonnegative square root of 25.
Since 5*5 = 25, √25 = 5
b.) √100
The symbol √100 represents the nonnegative square root of 81.
Since 10* 10 = 10, √100 = 10
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Find each square root.
a.) √81
b.) √144
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7. Answer : Perimeter is 40 inches
The area of a square is 100 square inches. Find its perimeter.
Area = 100 in2
First find the length of each side.
A = s2
100 = s2
Replace A with 100.
Both 10 and -10 , when multiplied by themselves are 100. However, since length can not be negative, s must be 10.
Hence, the length of each side is 10inches.
Now find the perimeter.
P = 4 . s
P =
P = 40
Replace s with 10.
Answer : Perimeter is 40 inches
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8. List some squares of numbers that are close to 125.
Numbers like 25, 49 and 64 are called perfect squares because when you take the square root then you get an answer that is whole number.
What if the number is not perfect square? How do you find the square root of a number like 125?
List some squares of numbers that are close to 125.
102= = = 144
The number 125 is not a perfect square, that is , 125 has no square root that is a whole number.
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Continued on the next slide.

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We know that 125 is greater than 121 or 112 and less than 144 or 122. So, the square root of 125 should be greater than 11 and less than 12.
√121 < √125 < √144
√ 112 < √125 < √ 122
11 < √125 < 12
Since 125 is close to 121 than to 144, the best whole number estimate for √125 would be 11.
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10. Irrational Numbers
Irrational Numbers: An irrational number is any real number that can not be expressed as a, b
where a and b are integers and b is not equal to 0.
Informally, this means numbers that cannot be represented as simple fractions. It can be deduced that they also cannot be represented as terminating or repeating decimals,
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11. get close but are not right.
Example: Pi is an irrational number. The value of Pi is (and more...)
There is no pattern to the decimals, and you cannot write down a simple fraction that equals Pi.
Values like 22 = 7
get close but are not right.
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12. Determine whether the number is rational or irrational. 0.22222...
The three dots means that the 2s keep repeating . This is repeating decimal, so it can be expressed as a fraction.
= 2 9
Thus, it is a rational number.
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The sets of real numbers includes both the rational numbers and irrational numbers.
Real numbers
Rational numbers
Integers
Whole
numbers
Irrational numbers
Rational number are numbers that can be expressed as a ratio of two integers, where the divisor are not zero.
Irrational number are numbers that can be named by non terminating , non repeating decimals.
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14. Solve the equation. Round decimal answer to the nearest tenth. y2 = 75
Solution: y2 = 75
y = √75 or - √75 Take the square root of each side.
√ 75 =
y ≈ 8.7 or y ≈ -8.7
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BREAK
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Assignments
Find each square root.
√121
√25
√169
√225
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18. √79 Find the best integer estimate for each square root √14
7) The area of a square is 169 square inches. Find its perimeter.
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19. Determine whether the numbers are rational or irrational. 0.83
Solve the equation. Round decimal answer to the nearest tenth.
a2 = 36
b2 = 14
c2 = 132
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20. Square Roots and Irrational Numbers Definition of square root
Very Good!
Let's Review
Today we will discuss about
Square Roots and Irrational Numbers
Definition of square root
In words: The square root of a number is one of its two equal factors.
In Symbols: If x2 = y, then x is a square root of y.
For example: The square root of 49 is 7 since 7.7 or 72 is 49. It is also true that -7.(-7) = 49. So -7 is another square root of 49.
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21. Review
The Square Root Symbol
The symbol √, called the radical sign is used to indicate a non-negative square root.
√49 indicates the non-negative square root of 49.
√49 = 7
-√49 indicates the negative square root of 49.
-√49 = -7
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Review
Find each square root.
a.) √25
The symbol √25 represents the nonnegative square root of 25.
Since 5*5 = 25, √25 = 5
b.) √100
The symbol √100 represents the nonnegative square root of 81.
Since 10* 10 = 10, √100 = 10
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23. Answer : Perimeter is 40 inches
Review
The area of a square is 100 square inches. Find its perimeter.
Area = 100 in2
First find the length of each side.
A = s2
100 = s2
Replace A with 100.
Both 10 and -10 , when multiplied by themselves are 100. However, since length can not be negative, s must be 10.
Hence, the length of each side is 10inches.
Now find the perimeter.
P = 4 . s
P =
P = 40
Replace s with 10.
Answer : Perimeter is 40 inches
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24. List some squares of numbers that are close to 125.
Review
Numbers like 25, 49 and 64 are called perfect squares because when you take the square root then you get an answer that is whole number.
What if the number is not perfect square? How do you find the square root of a number like 125?
List some squares of numbers that are close to 125.
102= = = 144
The number 125 is not a perfect square, that is , 125 has no square root that is a whole number.
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Continued on the next slide.

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Review
We know that 125 is greater than 121 or 112 and less than 144 or 122. So, the square root of 125 should be greater than 11 and less than 12.
√121 < √125 < √144
√ 112 < √125 < √ 122
11 < √125 < 12
Since 125 is close to 121 than to 144, the best whole number estimate for √125 would be 11.
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26. Review
Irrational Numbers
Irrational Numbers: An irrational number is any real number that can not be expressed as a, b
where a and b are integers and b is not equal to 0.
Informally, this means numbers that cannot be represented as simple fractions. It can be deduced that they also cannot be represented as terminating or repeating decimals,
Determine whether the number is rational or irrational.
The three dots means that the 2s keep repeating . This is repeating decimal, so it can be expressed as a fraction.
= 2 9
Thus, it is a rational number.
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Review
The sets of real numbers includes both the rational numbers and irrational numbers.
Real numbers
Rational numbers
Integers
Whole
numbers
Irrational numbers
Rational number are numbers that can be expressed as a ratio of two integers, where the divisor are not zero.
Irrational number are numbers that can be named by non terminating , non repeating decimals.
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28. Solve the equation. Round decimal answer to the nearest tenth. y2 = 75
Solution: y2 = 75
y = √75 or - √75 Take the square root of each side.
√ 75 =
y ≈ 8.7 or y ≈ -8.7
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Great Job!
Remember to do the practice worksheets!!!
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