BY: NIKI SADAT AFJEH AND ROBIN S.
BLOCK G
CHAPTER 6: SQUARE ROOTS
MAIN IDEAS AND CONCEPTS
6.1 SQUARE ROOTS
To find a number that has been squared is called finding the square root of the number. The
symbol, √ is used to signify a square root. The word “square” of a number is the product of that
number and itself. When a number has been squared, it has been increased to a second power.
For example, let’s look at the expression 6². The exponent “²”shows that the base which is 6 is
the number that is being squared.
Forms of writing exponents: 5 to the power of 3 or 5 to the third power will appear as 5³ which is
written in the exponential form. 5x5x5 is the factored form. One hundred and twenty five/ 125
would then be the standard form, which is the most common way a number is written in.
FUN FACT! 5² can be read as 5 squared and 5³ can be read as 5 cubed.
Further information on square roots...
The square root of a number is the value that, when it is multiplied by itself gives the original
number. Squaring a number is opposite to finding a square root of a number. When you square a
number, a new product is found, and taking the square root of the new product gives you the
original number.
Some numbers are perfect squares and some are not. A perfect square number is a number whose
square root is a whole number.
To check if a number is a perfect square, you can use factor trees!
For e.g.:
*http://qimg.icoachmath.com/qs/1-25000/8540_12145.gif
Each side is equal, 2x2x2x2=16 and 2x2x2x2=16 and so 16x16=256. Two hundred and fifty six
(256) is a perfect square!
Some examples of perfect square numbers
are:0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,etc...
6.2 MODELLING SQUARE ROOTS
Square roots can be used in everyday lives! They are needed when you have to find the whole
number that, when multiplied by itself, equals a new value, such as finding the length of a
square.
For example: A square of length “x” (unknown) has an area of 16 mm ². The area of the square is
Area= x², so x must be 4 mm since 4x4= 16, or in other words the square root of 16/ √16=4.
6.3 NON PERFECT SQUARE ROOTS
The majority of numbers are non-perfect square root numbers rather than perfect square root
numbers. A non-perfect square root number is not a whole number or a rational number; instead,
it is called an irrational number. If you use a calculator to find the square root of an irrational
number, the answer will only be part of a real answer because the decimal continues forever.
An alternative and easy way to find the square root of an irrational number can be determined by
finding out the two perfect square numbers it is between. For example: √30. The two perfect
squares that it is between are 25 and 36. Therefore, √30 must be in between √25 and √36, also
known as 5x5 and 6x6.
How to: Find the decimal values of non perfect roots
1st step: Find the two perfect square values that are above and below the non perfect square root.
The decimal value will be in between the whole number values of the perfect squares.
2nd step: Choose decimal values that when squared are just a tiny bit above and below the value
of the non-perfect square number. The actual value will be somewhere in between the two
squared numbers.
6.4/6.5 THE PYTHAGOREAN THEOREM/ APPLICATIONS OF THE PYTHAGOREAN
THEOREM
The Pythagorean Theorem is one of the most common used formulas in the world of
mathematics! It says that the perpendicular sides also known as the legs of a right triangle is
equal to the hypotenuse of the triangle, which is the longest side.
An easy way to think of this is using the formula, a² + b²= c² or c²= a²+b²
The “a” and b” are the legs and “c” is the hypotenuse. The square of the length of the hypotenuse
is equal to the sum of the squares of the lengths of the two perpendicular sides/legs of a right
triangle.
http://ncalculators.com/images/pythagoras-theorem.gif
FUN FACT! Right angles and the Pythagorean Theorem play an important part in solving many
applied problems.
REQUIRED MATH SKILLS
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You will need to have mastered the basic math skills such as; Multiplication, division,
addition, subtraction, estimating, rounding, etc...
You will need to know those things in order to figure out the perfect and non perfect
square numbers
You will also need to know the value of numbers
You will need to have memorized and mastered in how to use the Pythagorean Theorem
(a²+b²=c²) depending on the known values of the triangle legs
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You will need to know how to calculate square roots of numbers which requires knowing
the process and math skills of calculating square roots
Lastly, but not least you will need to have completely mastered the Pythagorean Theorem
so you will be capable of doing applied problems or applications of the Pythagorean
Theorem
TEST YOUR KNOWLEDGE! (QUESTIONS)
Q: √81
A: 81
/\
99
/\/\
3333
3X3=9 3X3=9 9X9=81
√81=9
Q: √144
A: 144
/\
12 12
/\ /\
3434
/\
/\
22 22
3x3x2x2x2x2= 144
√144=12
Q: √45
A: √45 is a NON PERFECT SQUARE NUMBER.
√45 is in between √36 and √49, the whole number would be in between 6 and 7.
√45 is closer to √47, which is 7x7. Therefore, a good estimate would be 6.7 since it is closer to 7.
6.7x6.7= 44.89 which is very close to 45.
Q: √96
A: √96 is a NON PERFECT SQUARE NUMBER.
√96 is in between √81 and √100; the whole number would be in between 9 and 10.
√96 is closer to √100, which is 10x10. Therefore, a good estimate would be 9.8 since it is closer
to 10.
9.9x9.8= 96.04 which is very close to 96.
Q: Calculate the length of one side of a square with an area of 54 cm².
A: A square root is the number that must be multiplied by itself to make another number. For a
square, the two sides are equal in length so you know that L x L = Area.
Since your area is 54, then L x L = 54.
The next step is to find the square root of 54, and we know for sure that just by looking at it, 54
is NOT A PERFECT SQUARE NUMBER.√54 is in between √49 and √64, the whole number
would then be in between 7 and 8. √54 is closer to the √49, which is 7x7. Therefore, a good
estimate would be 7.3 since it is closer to 7.
7.3x7.3=53.29 which is very close to 54. (*NOTE: You could have also used 7.4 as an estimate
but 7.4x7.4=54.76 which is a 0.76 difference from 54, but 7.3x7.3=53.29 which is only a 0.71
difference from 54, a much better estimate.)