Download - Kareköklü sayilar
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-Yukarıda noktalı kâğıt üzerinde oluşturulan kare modellerinin alanlarını ve kenar
uzunluklarını bulunuz.
-Bir kenar uzunluğu tam sayı olan karelerin alanları olabilecek sayıları da siz bulunuz.
SORU
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- Alanı 144 birimkare olan bir karenin bir kenarının uzunluğu bulunurken
kendimize
“Hangi sayının kendisiyle çarpımı 144 eder?” diye sorarız.
-Bu sorunun yanıtı 12 . 12 ya da (–12) . (–12) dir.
-Karenin bir kenar uzunluğu negatif bir sayı olamayacağı için 12 . 12 işlemini
düşünerek kenar uzunluğunu 12 birim olarak alırız.
•Karekök nedir?Sorusunun cevabını artık verebiliriz.
•Karenin alanı verildiğinde bir kenarı bulmak için yapılan işleme karekök alma
işlemi denir.
İlk mat.net
İlk mat.net
BİLGİ
BİLGİ
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-Aşağıdaki sayıların kareköklerini bulunuz.
√1 = √ 12 = 1’dir.
√ 0 = √ 02 = 0’dır.
√ 4 = √ 22 = 2’dir..
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Karekökü 1 ile 20 arasındaki sayılar
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ÖRNEK
ÖRNEK
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ÖRNEK
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ÖRNEK
ÖRNEK
İlk mat.net
Alanı 81 birim kare olan bir karesel bölgenin çevre uzunluğu kaç birimdir?
A) 32 B) 36 C) 40 D) 44
Aşağıdakilerden hangisi bir karesel sayıdır?
A) 6 B) 12 C) 24 D) 36
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ÖRNEKLER
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ÖRNEKLER
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ÖRNEK
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ÖNEMLİBİLGİ
• Negatif sayılıların karekökünü alamayız.
• ‘’NEYİN KARESİ -4 ‘’hiç bir zaman bir a sayısının karesi – olamaz
a.a=a2≠-4
• √-4 sayısının karekökünü alalım.
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ÖRNEKLER
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Tam Kare Olmayan Sayıların
Kareköklerini Tahmin Etme
• Bu işlemi bir örnekle adım adım anlatalım.
• √164 sayısının yaklaşık değerini bulalım.
• 164 sayısının tam kare olmadığı öğrenmiştik.
• 164 sayısına en yakın iki tam kare sayı bulalım.
• Bunlar 144 ve 169 olur.164’ü bu sayılar arasına yazıp karekökünü
alalım.
• 144< 164< 169
• √144< √164< √169
• 12 < √164 < 13
• Böylece √164 ün 12 ile 13 arasında bir sayı olduğunu anlayabiliriz.
• √164 = 12,…….dir.ŞİMDİDE VİRGÜLDEN SONRASINI
TAHMİN EDELİM.
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• 164 sayısı 144 de mi ? 169 daha yakın?
• Bir rasyonel sayıyı sayı doğrusunda gösterirken kullandığımız yöntemi
kullanacağız.
• 169-144=25 tir.bir bütün 25 parçaya bölünmüş 20 si alınmış gibi
düşünelim.
12 8,1225
2012
8,025
20• 12 nin virgülden sonrası
144 164 169
20 br 5 br
√144 √164 √169
20 br 5 br
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ÖRNEKLER
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√300 sayısının yaklaşık değeri kaçtır?
ÖRNEK
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ÖRNEKLER
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KAREKÖK İÇİNDEKİ BİR SAYIYI
ŞEKLİNDE YAZMA
dir.2 baba
ba
48
300
192
Sayılar büyüdükçe karesel sayı çarpanlarını bulmakzorlaşmaktadır. Böyle durumlarda karekök içindeki sayı asalçarpanlarına ayrılır.
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ÖRNEK
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ÖRNEK
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İlk mat.net
ÖRNEK
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ÖRNEK
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ÖRNEK
ÖRNEK
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dir.2baba
ba
510
39
23
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ÖRNEKLER
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ÖRNEKLER
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ÖRNEK
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ÖRNEK
ÖRNEK
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ÖRNEK
ÖRNEK
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Kareköklü sayılarla toplama ve çıkarma işlemlerinin yapılabilmesi için karekök
içindeki sayıların aynı olması gerekir.
ÖRNEK
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ÖRNEK
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ÖRNEK
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ÖRNEK
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ÖRNEK
ÖRNEK
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ÖRNEK
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ÖRNEKLER
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ÖRNEK
ÖRNEK
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a a . a
ÖRNEK
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İlk mat.net
Yukarıda verilen kare ve dikdörtgensel bölgelerin alanlarını bulunuz.
ÖRNEK
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ÖRNEK
ÖRNEK
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23.3
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1 5 . 15
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22
2
2
b– a b) (a b)– (a
b– a )b (a )b– (a
b– a b) a( b)– a(
b– a )b a( )b– a(
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)3 2( )3– 2( 1.
2. 1) 3( 1)– 3(
3. a) (3 a)– (3
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yazilir. olarak b
a
b
a
Kareköklü sayılarla bölme işlemi yapılırken; karekök içindeki
sayılar ortak karekök içine alınarak bölünür. Karekök dışındaki
sayılar kendi aralarında bölünebiliyorsa bölünüp karekök dışına
yazılır.
100
49
81
1
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1.
2.
3.
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Paydasında kareköklü bir sayı bulunan ifadenin paydasını rasyonel sayı
yapmak için ifadeyi paydasındaki kareköklü sayı ile genişletiriz.
Paydasında √a – √b şeklinde bir ifade bulunan sayının paydasını rasyonel
sayı yapmak için ifade √a + √b (eşlenik) ile genişletilir.
Paydada √a + √b varsa rasyonel ifade bu kez √a – b ile genişletilir.
BİLGİ
25
53
25
53
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1. YOL 2. YOL
Ondalık kesirlerin karekökünü bulmak
için önce ondalık kesirlerin kesir gösterimi
yazılır. Sonra bu kesrin pay ve paydasının
ayrı ayrı karekökleri alınarak sonuca ulaşılır.
5,010
5
100
250,25
Sayıyı virgülsüz gibi düşünüp
kök dışına çıkaralım.
50,25
Sayının virgülden sonraki basamak
sayısının yarısı kadar kök dışına
çıkardığımız sayının virgülü sola
kaydırılır.
5,00,25
Virgülden sonra 2 basamak vardı
bu nedenle 5 sayısının virgülü bir
basamak sola kaydırıldı
1,110
11
100
1211,21
İlk mat.net
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İlk mat.net
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İlk mat.net
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İlk mat.net
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İlk mat.net
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İlk mat.net
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SORU- 1
27 birim karoya en az kaç karo daha eklersek bir kare oluşturabiliriz.
A) 2 B)7 C)9 D)11
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SORU- 2
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SORU- 3
16 5 17
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SORU- 4
? 45
180
108
75
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SORU- 5
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SORU- 6
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SORU- 7
0,49
1 . 01,0
1
1
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SORU- 8
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SORU- 9
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SORU- 10
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SORU- 11
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SORU- 12
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SORU- 13
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SORU- 14
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SORU- 15
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SORU- 16
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SORU- 17
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SORU- 18
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SORU- 19
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SORU- 20
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SORU- 21
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