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Jennifer Lorance
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তারিখঃ 2022-08-22T14:26:53-06:00 ক্যাটাগরিঃ

The easy expansion of (a+b)^3 – a plus b whole cube

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    aalan

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    1. Nusaiba
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      নতুন

      At first we need to understand (a + b)3 formula in detail in the following section.

      The (a + b)^3 Formula

      To find the cube of a binomial, we will just multiply (a + b)(a + b)(a + b). (a + b)3 formula is also an identity. It holds true for every value of a and b.The (a + b)3 is given as,

      (a + b)3 = (a + b)(a + b)(a + b)

      = (a2 + 2ab + b2)(a + b)

      = a3 + a2b + 2a2b + 2ab2 + ab2 + b3

      = a3 + 3a2b + 3ab2 + b3

      = a3 + 3ab(a+b) + b3

      Therefore, (a + b)^3 formula is:

      (a + b)3 = a3 + 3a2b + 3ab2 + b3

      Examples (a + b)3 Formula

      Example : Solve the following expression using suitable algebraic identity: (2x + 3y)3

      Solution:

      To find: (2x + 3y)3
      Using (a + b)3 Formula,
      (a + b)3 = a3 + 3a2b + 3ab2 + b3
      = (2x)3 + 3 × (2x)2 × 3y + 3 × (2x) × (3y)2 + (3y)3
      = 8x3 + 36x2y + 54xy2 + 27y3

      Answer: (2x + 3y)3 = 8x3 + 36x2y + 54xy2 + 27y3

      Example : Find the value of x3 + 8y3 if x + 2y = 6 and xy = 2.

      Solution:

      To find: x3 + 8y3
      Given: x + 2y = 6
      xy = 2
      Using (a + b)3 formula,
      (a + b)3 = a3 + 3a2b + 3ab2 + b3
      Here, a = x; b = 2y
      Therefore,
      (x + 2y)3 = x3 + 3 × x2 × (2y) + 3 × x × (2y)2 + (2y)3
      (x + 2y)3 = x3 + 6x2y + 12xy2 + 8y3
      63 = x3 + 6xy(x + 2y) + 8y3
      216 = x3 + 6 × 2 × 6 + 8y3
      x3 + 8y3 = 144

      Answer:  x3 + 8y3 = 144

      Example : Solve the following expression using (a + b)3 formula:

      (5x + 2y)3

      Solution:

      To find: (5x + 2y)3
      Using (a + b)3 Formula,
      (a + b)3 = a3 + 3a2b + 3ab2 + b3
      = (5x)3 + 3 × (5x)2 × 2y + 3 × (5x) × (2y)2 + (2y)3
      = 125x3 + 150x2y + 60xy2 + 8y3

      Answer: (5x + 2y)3  = 125x3 + 150x2y + 60xy2 + 8y3

      Expansion of (a + b)3 Formula

      (a + b)3 formula is read as a plus b whole cube. Its expansion is expressed as (a + b)3 = a3 + 3a2b + 3ab2 + b3

      Simplify Numbers Using the (a + b)3 Formula

      Let us understand the use of the (a + b)3 formula with the help of the following example.
      Example: Find the value of (20 + 5)3 using the (a + b)3 formula.
      To find: (20 + 5)3
      Let us assume that a = 20 and b = 5.
      We will substitute these in the formula of (a + b)3.
      (a + b)3 = a3 + 3a2b + 3ab2 + b3
      (20+5)3 = 203 + 3(20)2(5) + 3(20)(5)2 + 53
      =8000 + 6000 + 1500 + 125
      = 15625
      Answer: (20 + 5)3 = 15625.

      Using the (a + b)3 Formula Steps

      The following steps are followed while using (a + b)3 formula.

      • Firstly observe the pattern of the numbers whether the numbers have whole ^3 as power or not.
      • Write down the formula of (a + b)3
      • (a + b)3 = a3 + 3a2b + 3ab2 + b3
      • Substitute the values of a and b in the (a + b)3 formula and simplify.

      Thanks!

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