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Factories this further:\newline(10x2−24xy+16y2)(8x2−24xy+16y2)(10x^2-24xy+16y^2) (8x^2-24xy+16y^2)

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Q. Factories this further:\newline(10x2−24xy+16y2)(8x2−24xy+16y2)(10x^2-24xy+16y^2) (8x^2-24xy+16y^2)
  1. Recognize Perfect Squares: Recognize that both quadratic expressions are perfect squares. The first expression, 10x2−24xy+16y210x^2-24xy+16y^2, can be factored into (ax−by)2(ax - by)^2 where aa and bb are numbers that when squared give 10x210x^2 and 16y216y^2 respectively, and 2ab2ab gives 24xy24xy. Similarly, the second expression, 8x2−24xy+16y28x^2-24xy+16y^2, can be factored into (cx−dy)2(cx - dy)^2 where (ax−by)2(ax - by)^200 and (ax−by)2(ax - by)^211 are numbers that when squared give (ax−by)2(ax - by)^222 and 16y216y^2 respectively, and (ax−by)2(ax - by)^244 gives 24xy24xy.
  2. Find Values of a, b, c, d: Find the values of aa, bb, cc, and dd. For the first expression, we need two numbers whose product is 1010 and whose square gives 1616 when multiplied. These numbers are 22 and 44, so a=10a = \sqrt{10} and b=4b = 4. For the second expression, we need two numbers whose product is bb00 and whose square gives 1616 when multiplied. These numbers are 22 and 44, so bb44 and bb55.
  3. Write Factored Forms: Write the factored form of each expression.\newlineThe first expression becomes (a10x−4y)2=(10x−4y)2(a\sqrt{10}x - 4y)^2 = (\sqrt{10}x - 4y)^2.\newlineThe second expression becomes (c2x−4y)2=(22x−4y)2(c\sqrt{2}x - 4y)^2 = (2\sqrt{2}x - 4y)^2.
  4. Multiply Factored Forms: Multiply the factored forms of the two expressions.\newlineNow we multiply the two factored forms: (10x−4y)2×(22x−4y)2(\sqrt{10}x - 4y)^2 \times (2\sqrt{2}x - 4y)^2.
  5. Recognize Square of a Product: Recognize that the multiplication of two squares can be expressed as the square of a product.\newlineWe can write the multiplication as: [(10x−4y)⋅(22x−4y)]2 [ (\sqrt{10}x - 4y) \cdot (2\sqrt{2}x - 4y) ]^2 .
  6. Multiply Expressions Inside Brackets: Multiply the expressions inside the brackets.\newlineNow we need to multiply (10x−4y)(\sqrt{10}x - 4y) by (22x−4y)(2\sqrt{2}x - 4y).\newlineFirst, multiply 10x\sqrt{10}x by 22x2\sqrt{2}x to get 220x22\sqrt{20}x^2.\newlineSecond, multiply 10x\sqrt{10}x by −4y-4y to get −410xy-4\sqrt{10}xy.\newlineThird, multiply −4y-4y by 22x2\sqrt{2}x to get (22x−4y)(2\sqrt{2}x - 4y)00.\newlineFourth, multiply −4y-4y by −4y-4y to get (22x−4y)(2\sqrt{2}x - 4y)33.
  7. Combine Like Terms: Combine like terms. Combine −410xy-4\sqrt{10}xy and −82xy-8\sqrt{2}xy. Since 10\sqrt{10} and 2\sqrt{2} are not like terms, we cannot combine these two terms. So, the expression inside the brackets is 220x2−410xy−82xy+16y22\sqrt{20}x^2 - 4\sqrt{10}xy - 8\sqrt{2}xy + 16y^2.
  8. Write Final Factored Form: Write the final factored form.\newlineThe final factored form is [220x2−410xy−82xy+16y2]2[2\sqrt{20}x^2 - 4\sqrt{10}xy - 8\sqrt{2}xy + 16y^2]^2.

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