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Cubes and Cube roots

Η. Κ. Σ. Ι. Ε. Ζ. Ω. Φ. Cubes and Cube roots. Γ. Θ. Β. Α. θ. Δ. Contents. Introduction Perfect Cubes Cube root Cube root by prime factorisation Cube root by estimation. introduction.

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Cubes and Cube roots

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  1. Η Κ • Σ Ι Ε Ζ Ω Φ Cubes and Cube roots Γ Θ Β Α • θ Δ

  2. Contents • Introduction • Perfect Cubes • Cube root • Cube root by prime factorisation • Cube root by estimation.

  3. introduction • one of the great mathematical geniuses, S ramanujanhad a visit of prof G H hardy. He came with a taxi whose number is 1729 .he described the number as a dull number. Ramanujan quickly pointed that 1729 was indeed interesting. He said it was the smallest number that can be expressed as the sum of two cubes in two different ways. 1729 = 1728+1=123 + 13 1729 = 1000+729=103 + 93

  4. What is a cube Numbers like 1, 8 ,27… are called cube numbers or a perfect cube . We get perfect cubes by multiplying a number 3 times with the same number. A 3

  5. For example- 1×1×1 = 13or 1cube, 8= 23 , 27= 33…etc.

  6. CUBE ROOTS • Finding the square root, as you know , is the inverse operation of squaring. Similarly finding the cube root is the inverse operation of finding cube. • We know that 23= 8; so we say that cube root of 8 is 2. We write = 2. • 83=512;so of =8. THE SYMBOL “ ” DENOTES “CUBE ROOT ”

  7. Cube Root Through Factorisation Method CONSIDER “3375”: 3375=3×3×3×5×5×5 ‾‾‾‾‾ ‾‾‾‾‾ =33 × 53=(3×5)3 = 15 We find its cube root by prime factorisation. The factors are ; Therefore, cube root of 3375= 15

  8. EXAMPLE ; Find the cube root of 8000. ANSWER: Prime factorisation of 8000 is 8000=2×2×2×2×2×2×5×5×5 Therefore “ ” = 2×2×5=20

  9. CUBE ROOT BY ESTIMATIONMETHOD To find the cube root of a cube number, the following method can be used. • STEP 1. 857375= 857 375 ↓ ↓ second number first number We get 375 & 857 as two groups of three digits each Take a cube number; 857375. Make group of three digits starting from the right most digit of the number.

  10. STEP 2 375 ‾ So,we get 5 at the unit’s place cube root. • STEP 3 857 We know that 9^3=729 & 10^3=1000.Also, 729<857<1000.We take the one”splace,place the as the ten “s place of the required cube root.So,we get “CR”857375=95 First group i.e.,375 will give the one’s digit of the required cube root. The number 375 ends with 5.We know that 5 comes at the unit’s place of a number only when it’s cube root ends in 5. Now we take the next group

  11. WORK SHEET (FOR FA-3) Find the cube root of each of the of the following by prime factorization method. • 64 • 512 Find the cube root through estimation • i.17576 • ii. 3375 • iii.1331 ANSWER • i. 3√64=2×2×2×2×2×2=2×2=4 ‾‾‾‾‾ ‾‾‾‾‾ • ii.3√512=2×2×2×2×2×2×2×2×2 =2×2×2=8 ‾‾‾‾‾ ‾‾‾‾‾ ‾‾‾‾‾ • iii.3√10648=11×11×11×2×2×2=11×2=22 ‾‾‾‾‾‾‾ ‾‾‾‾‾ • i. 26 • ii 15 • iii 11

  12. WHICH OF THE FOLLOWING ARE NOT PERFECT CUBE • 216 • 128 • 1000 ANS – 128 IS NOT A PERFECT CUBE • IS 68600 A PERFECT CUBE ? IF NOT FIND THE SMALLEST NUMBER BY WHICH IT SHOULD BE MULTIPLIED TO GET A PERFECT CUBE. ANS – NO IT’S NOT A PERFECT CUBE. IT SHOULD BE MULTIPLIED BY 5

  13. IS 1188 A PERFECT CUBE ?IF NOT, BY WHICH SMALLEST NATURAL NUMBER SHOULD IT BE DIVIDED SO THAT THE QUOTIENT IS A PERFECT CUBE. ANS –NO IT’S NOT A PERFECT CUBE , IT SHOULD BE DIVIDED BY 44 TO GET A PERFECT CUBE.

  14. Thank you

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