Simplification


  1. The sum of the cubes of two numbers is 793. The sum of the numbers is 13. Then the difference of the two numbers is









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    Let the numbers be a and b where a > b.
    According to the question,
    a3 + b3 = 793
    and a + b = 13
    ∴  (a + b)3 = a3 + b3 + 3ab (a + b)
    ⇒  (13)3 = 793 + 3ab ×13
    ⇒  2197 = 793 + 39ab
    ⇒  39ab = 2197 – 793 = 1404

    ⇒  ab =
    1404
    = 36
    39

    ∴  (a + b)2 = (a + b)2 – 4ab
    = (3)2 = 4 × 36
    = 169 – 144 = 25
    ⇒  a – b = √25 = 5

    Correct Option: C

    Let the numbers be a and b where a > b.
    According to the question,
    a3 + b3 = 793
    and a + b = 13
    ∴  (a + b)3 = a3 + b3 + 3ab (a + b)
    ⇒  (13)3 = 793 + 3ab ×13
    ⇒  2197 = 793 + 39ab
    ⇒  39ab = 2197 – 793 = 1404

    ⇒  ab =
    1404
    = 36
    39

    ∴  (a + b)2 = (a + b)2 – 4ab
    = (3)2 = 4 × 36
    = 169 – 144 = 25
    ⇒  a – b = √25 = 5


  1. What is the smallest number by which 625 must be divided so that the quotient is a perfect cube ?









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    ∴  625 = 5 × 5 × 5 × 5 = 53 × 5
    For the smallest cube number,
    625 should be divided 5,
    625 ÷ 5 = 125 = 53

    Correct Option: B


    ∴  625 = 5 × 5 × 5 × 5 = 53 × 5
    For the smallest cube number,
    625 should be divided 5,
    625 ÷ 5 = 125 = 53



  1. The value of (1001)3 is









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    Look at the pattern :
    1001 × 1001 = 1002001
    1001×1001 × 1001 = 1003003001

    Correct Option: A

    Look at the pattern :
    1001 × 1001 = 1002001
    1001×1001 × 1001 = 1003003001


  1. If   x = √3 + √2  then the value of   x3
    1
      is
    x3









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    x = √3 + √2

    ∴ 
    1
    =
    1
    x3 + √2

    =
    3 − √2
    (√3 + √2)(√3 − √2)

    =
    3 − √2
    = √3 − √2
    3 − 2

    ∴  x −
    1
    = √3 + √2 − √3 + √2 = 2√2
    x

    ∴  x3
    1
    = x −
    1
    3 + 3 x −
    1
    x3xx

    = (2√2)3 + 3 × 2√2
    = 16√2 + 6√2 = 22√2

    Correct Option: C

    x = √3 + √2

    ∴ 
    1
    =
    1
    x3 + √2

    =
    3 − √2
    (√3 + √2)(√3 − √2)

    =
    3 − √2
    = √3 − √2
    3 − 2

    ∴  x −
    1
    = √3 + √2 − √3 + √2 = 2√2
    x

    ∴  x3
    1
    = x −
    1
    3 + 3 x −
    1
    x3xx

    = (2√2)3 + 3 × 2√2
    = 16√2 + 6√2 = 22√2



  1. If the square root of x is the cube root of y, then the relation between x and y is









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    x = 3y
    ⇒  x1/2 = y1/3
    ⇒  (x1/2)6 = (y1/3)6
    ⇒  x3 = y2

    Correct Option: A

    x = 3y
    ⇒  x1/2 = y1/3
    ⇒  (x1/2)6 = (y1/3)6
    ⇒  x3 = y2