Squares & Square Roots. Perfect Squares

Transcription

1 Squares & Square Roots Perfect Squares

2 Square Number Also called a perfect square A number that is the square of a whole number Can be represented by arranging objects in a square.

3 Square Numbers

4 1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 Square Numbers

5 1 x 1 = 1 2 x 2 = 4 3 x 3 = 9 4 x 4 = 16 Activity: Calculate the perfect squares up to 15 2 Square Numbers

6 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 = x 10 = x 11 = x 12 = x 13 = x 14 = x 15 = 225

7 Activity: Identify the following numbers as perfect squares or not. i. 16 ii. 15 iii. 146 iv. 300 v. 324 vi. 729

8 Activity: Identify the following numbers as perfect squares or not. i. 16 = 4 x 4 ii. 15 iii. 146 iv. 300 v. 324 = 18 x 18 vi. 729 = 27 x 27

9 Activity: Graph the perfect squares between 1 and 100 on a number line

10 Activity: Graph the perfect squares between 1 and 100 on a number line

11 Square Numbers 4cm 4cm 16 cm 2 One 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.

12 Square Numbers 4cm 4cm 16 cm 2 The large square has an area of 4cm x 4cm = 16 cm 2. The number 4 is called the square root of 16. We write: 4 = 16

13 Square Root A number which, when multiplied by itself, results in another number. Ex: 5 is the square root of = 25

14 Exponents In 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. Factors Product of Factors Description Translation 2 x 2 = 4 2 is a factor 2 times 4 is the 2 nd power of 2 2 x 2 x 2 = 8 2 is a factor 3 times 8 is the 3 rd power of 2 2 x 2 x 2 x 2 = 16 2 is a factor 4 times 16 is the 4 th power of 2 2 x 2 x 2 x 2 x 2 = 32 2 is a factor 5 times 32 is the 5 th 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 = is a factor 7 times 128 is the 7 th power of 2 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = is a factor 8 times 256 is the 8 th 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. Base Exponent 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: 2 1,000,000 read as two raised to the millionth power 2 n read as two raised to the nth power.

15 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 = raised to the fifth power Let us take another look at the table from above to see how exponents work. Exponential Form Factor Form Standard Form 2 2 = 2 x 2 = = 2 x 2 x 2 = = 2 x 2 x 2 x 2 = = 2 x 2 x 2 x 2 x 2 = = 2 x 2 x 2 x 2 x 2 x 2 = = 2 x 2 x 2 x 2 x 2 x 2 x 2 = = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256

16 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 = 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 = 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 = raised to the seventh power Example 5: Solution: Write 10 3, 3 6, and 1 8 in factor form and in standard form. Exponential Form Factor Form Standard Form x 10 x 10 1, x 3 x 3 x 3 x 3 x 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, = 10,000

17 Special Exponents The following rules apply to numbers with exponents of 0, 1, 2 and 3: Rule Example Any number (except 0) raised to the zero power is equal to = 1 Any number raised to the first power is always equal to itself. 8 1 = 8 If a number is raised to the second power, we say it is squared. 3 2 is read as three squared If a number is raised to the third power, we say it is cubed. 4 3 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. 2 4 = = = = = = = = = =

18 P E MD AS 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 exponents My,Dear: do all multiplications and divisions as they occur from left to right Aunt,Sally: do all additions and subtractions as they occur from left to right. Example: (8-6) 2 Expression in parentheses gets calculated first = (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. = (4) Next in order comes multiplication. Multiplication and Division always come before addition or division, even if to the right. = 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. = Now finally we can do the addition. = 30

19 Order of Operations with Square Roots Make sure you follow PEMDAS Treat the Radical Sign as a special grouping symbol. Example: Do what s inside the grouping symbols first. = - 6(9) + 5(1) = = -49