How can we express the equation 64x^6 + 27 as a sum of cubes?

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To express the equation 64x6 + 27 as a sum of cubes, we first recognize that this can be rewritten in the form of a3 + b3, where:

  • a corresponds to (4x2) since (4x2)3 = 64x6, and
  • b corresponds to 3 since 33 = 27.

Now we rewrite the expression:

64x6 + 27 = (4x2)3 + 33

Next, we can apply the sum of cubes formula, which is:

a3 + b3 = (a + b)(a2 - ab + b2)

Substituting a = 4x2 and b = 3 into the formula gives us:

(4x2 + 3)((4x2)2 - (4x2)(3) + 32)

Now simplifying:

  • (4x2)2 = 16x4
  • (4x2)(3) = 12x2
  • 32 = 9

Putting it all together, we get:

(4x2 + 3)(16x4 - 12x2 + 9)

Thus, we can express 64x6 + 27 as the sum of cubes:

Final Result:

64x6 + 27 = (4x2 + 3)(16x4 - 12x2 + 9)

This representation not only illustrates the polynomial’s structure but also showcases how sum of cubes can simplify complex expressions into more manageable forms.

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