Squares & Square Roots

Perfect Squares

Square NumberAlso called a “perfect square”

A number that is the square of a whole number

Can be represented by arranging objects in a square.

Square Numbers

Square Numbers

1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 =

16

Square Numbers

1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16Activity: Calculate the perfect squares up to 152…

Perfect Squares

1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 5 x 5 = 25 6 x 6 = 36 7 x 7 = 49 8 x 8 = 64

9 x 9 = 81 10 x 10 =

100 11 x 11 =

121 12 x 12 =

144 13 x 13 =

169 14 x 14 =

196 15 x 15 =

225

Activity:Identify the following

numbers as perfect squares or not.

i. 16ii. 15

iii. 146iv. 300v. 64

vi. 121

Activity:Identify the following

numbers as perfect squares or not.

i. 16 = 4 x 4ii. 15iii. 146iv. 300v. 64 = 8 x 8vi. 121 = 11 x 11

Square NumbersOne property of a

perfect square is that it can be represented by a

square array. Each small square in the

array shown has a side length of 1cm.

The large square has a side length of 4 cm.

4cm

4cm 16 cm2

Square Numbers

The large square has an area of 4cm x 4cm = 16

cm2.

The number 4 is called the square root of 16.

We write: 4 = 16

4cm

4cm 16 cm2

Square Root

A number which, when multiplied by itself, results

in another number.

Ex: 5 is the square root of 25.

5 = 25

ExponentsIn the table below, the number 2 is written as a factor repeatedly. The product of factors is also displayed in this table. Each product of factors is called a power.

FactorsProduct of Factors Description Translation

2 x 2 = 4 2 is a factor 2 times 4 is the 2nd power of 2

2 x 2 x 2 = 8 2 is a factor 3 times 8 is the 3rd power of 2

2 x 2 x 2 x 2 = 16 2 is a factor 4 times 16 is the 4th power of 2

2 x 2 x 2 x 2 x 2 = 32 2 is a factor 5 times 32 is the 5th power of 2

2 x 2 x 2 x 2 x 2 x 2 = 64 2 is a factor 6 times 64 is the 6 power of 2

2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 2 is a factor 7 times 128 is the 7th power of 2

2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 2 is a factor 8 times 256 is the 8th power of 2

Writing 2 as a factor one million times would be a very time-consuming and tedious task. A better way to approach this is to use exponents. Exponential notation is an easier way to write a number as a product of many factors.

BaseExponent The exponent tells us how many times the base is used as a factor.

For example, to write 2 as a factor one million times, the base is 2, and the exponent is 1,000,000. We write this number in exponential form as follows:

21,000,000 read as two raised to the millionth power

2n read as two raised to the nth power.

Example 1: Write 2 x 2 x 2 x 2 x 2 using exponents, then read your answer aloud.  

               

  

Solution: 2 x 2 x 2 x 2 x 2  =  25 “2 raised to the fifth power”

Let us take another look at the table from above to see how exponents work.

ExponentialForm

FactorForm

StandardForm

22 = 2 x 2 = 4

23 = 2 x 2 x 2 = 8

24 = 2 x 2 x 2 x 2 = 16

25 = 2 x 2 x 2 x 2 x 2 = 32

26 = 2 x 2 x 2 x 2 x 2 x 2 = 64

27 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128

28 =2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 =

256

So far we have only examined numbers with a base of 2. Let's look at some examples of writing exponents where the base is a number other than 2.

 Example 2: Write 3 x 3 x 3 x 3 using exponents, then read your answer aloud.

Solution: 3 x 3 x 3 x 3  =  34 3 raised to the fourth power

 

 Example 3: Write 6 x 6 x 6 x 6 x 6 using exponents, then read your answer aloud.

Solution: 6 x 6 x 6 x 6 x 6  =  65 6 raised to the fifth power

 Example 4: Write 8 x 8 x 8 x 8 x 8 x 8 x 8 using exponents, then read your answer aloud.

Solution: 8 x 8 x 8 x 8 x 8 x 8 x 8  =  87 8 raised to the seventh power

 

 

Example 5: Write 103, 36, and 18 in factor form and in standard form.

 

Solution: ExponentialForm

FactorForm

StandardForm

103 10 x 10 x 10 1,000

36 3 x 3 x 3 x 3 x 3 x 3 729

18 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 1

Notice in powers of 10, the exponent tells you how many zeroes come after the 1.102 = 100103 = 1,000104 = 10,000

So far we have only examined numbers with a base of 2. Let's look at some examples of writing exponents where the base is a number other than 2.

 P. 94 EXAMPLE 3 Expand each notation and find it’s value

a) 23

Solution:

This means to re-write the expression as a product of repeated factors. The exponent tells you how many times to repeat the factor (which is the base)

2 x 2 x 2 = 8

 

 

 

p.95 EX 6 Write 102, 103, and 104 in factor form and in standard form.

 

Another wording for “exponential form” is to use the word “power”.

100 is the 2nd power of 10, or 102

Exampe 7 b) Write 10,000,000 as a power of 10.How many zeroes? ___107

ExponentialForm

FactorForm

StandardForm

102 10 x 10 100

103 10 x 10 x 10 1,000

104 10 x 10 x 10x 10 10,000

Notice in powers of 10, the exponent tells you how many zeroes come after the 1.102 = 100103 = 1,000104 = 10,000

Special ExponentsThe following rules apply to numbers with exponents of  0, 1, 2 and 3:

RuleExample

Any number (except 0) raised to the zero power is equal to 1. 1490 = 1

Any number raised to the first power is always equal to itself. 81 = 8

If a number is raised to the second power, we say it is squared.

32 is read as three squared

If a number is raised to the third power, we say it is cubed. 43 is read as four cubed

Why does any nonzero number to the zero power equal 1? It just makes sense in the pattern of exponents.Each exponent that is one less than the previous one is the power divided by the base.

24 = 16 34 =

23 = 8 33 =

22 = 4 32 =

21= 2 31=

20= 1 30=

÷2

÷2

÷2

÷2

Order of Operatons

Please: do all operations within parentheses and other grouping symbols (such as [ ], or operations in numerators and denominators of fractions) from innermost outward.

Excuse: calculate exponentsMy,Dear: do all multiplications and divisions as they occur from left to rightAunt,Sally:  do all additions and subtractions as they occur from left to right.

Example: 20 – 2 + 3(8 - 6)2 Expression in parentheses gets calculated first

= 20 – 2 + 3(2)2 Next comes all items with exponents. The exponent only applies to the item directly to the left of it. In this case, only the (2) is squared.

= 20 – 2 + 3(4) Next in order comes multiplication. Multiplication and Division always come before addition or division, even if to the right.

= 20 – 2 + 12 Now when choosing between when to do addition and when to do subtraction, always go from left to right, so do 20-2 first, because the subtraction is to the left of the addition.

= 18 + 12 Now finally we can do the addition.

= 30

PEMDAS

Order of Operations with Square Roots

15816

:

Example

Make sure you follow PEMDAS Treat the Radical Sign as a special grouping symbol.

Do what’s inside the “grouping symbols” first.

= - 6(9) + 5(1)= -54 + 5= -49