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

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Identify independent and dependent events

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Q. Factories this further:

\newline

(10x^2-24xy+16y^2) (8x^2-24xy+16y^2)

Recognize Perfect Squares: Recognize that both quadratic expressions are perfect squares. The first expression, $10x^2-24xy+16y^2$ , can be factored into $(ax - by)^2$ where $a$ and $b$ are numbers that when squared give $10x^2$ and $16y^2$ respectively, and $2ab$ gives $24xy$ . Similarly, the second expression, $8x^2-24xy+16y^2$ , can be factored into $(cx - dy)^2$ where $(ax - by)^2$ $0$ and $(ax - by)^2$ $1$ are numbers that when squared give $(ax - by)^2$ $2$ and $16y^2$ respectively, and $(ax - by)^2$ $4$ gives $24xy$ .
Find Values of a, b, c, d: Find the values of $a$ , $b$ , $c$ , and $d$ . For the first expression, we need two numbers whose product is $10$ and whose square gives $16$ when multiplied. These numbers are $2$ and $4$ , so $a = \sqrt{10}$ and $b = 4$ . For the second expression, we need two numbers whose product is $b$ $0$ and whose square gives $16$ when multiplied. These numbers are $2$ and $4$ , so $b$ $4$ and $b$ $5$ .
Write Factored Forms: Write the factored form of each expression. $\newline$ The first expression becomes $(a\sqrt{10}x - 4y)^2 = (\sqrt{10}x - 4y)^2$ . $\newline$ The second expression becomes $(c\sqrt{2}x - 4y)^2 = (2\sqrt{2}x - 4y)^2$ .
Multiply Factored Forms: Multiply the factored forms of the two expressions. $\newline$ Now we multiply the two factored forms: $(\sqrt{10}x - 4y)^2 \times (2\sqrt{2}x - 4y)^2$ .
Recognize Square of a Product: Recognize that the multiplication of two squares can be expressed as the square of a product. $\newline$ We can write the multiplication as: $[ (\sqrt{10}x - 4y) \cdot (2\sqrt{2}x - 4y) ]^2$ .
Multiply Expressions Inside Brackets: Multiply the expressions inside the brackets. $\newline$ Now we need to multiply $(\sqrt{10}x - 4y)$ by $(2\sqrt{2}x - 4y)$ . $\newline$ First, multiply $\sqrt{10}x$ by $2\sqrt{2}x$ to get $2\sqrt{20}x^2$ . $\newline$ Second, multiply $\sqrt{10}x$ by $-4y$ to get $-4\sqrt{10}xy$ . $\newline$ Third, multiply $-4y$ by $2\sqrt{2}x$ to get $(2\sqrt{2}x - 4y)$ $0$ . $\newline$ Fourth, multiply $-4y$ by $-4y$ to get $(2\sqrt{2}x - 4y)$ $3$ .
Combine Like Terms: Combine like terms. Combine $-4\sqrt{10}xy$ and $-8\sqrt{2}xy$ . Since $\sqrt{10}$ and $\sqrt{2}$ are not like terms, we cannot combine these two terms. So, the expression inside the brackets is $2\sqrt{20}x^2 - 4\sqrt{10}xy - 8\sqrt{2}xy + 16y^2$ .
Write Final Factored Form: Write the final factored form. $\newline$ The final factored form is $[2\sqrt{20}x^2 - 4\sqrt{10}xy - 8\sqrt{2}xy + 16y^2]^2$ .

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