Mathematics: What identity can be used to rewrite the expression 8x^3+243

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What identity can be used to rewrite the expression

Ответ:

mkingston1705

14.03.2021

Mathematics

Sum of Cubes

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Ответ:

ghari112345

14.03.2022

Mathematics

To rewrite this expression we can use the sum of cubes identity: a^3+b^3=(a+b)(a^2-ab+b^2)

. Notice that we can express 8 as a cube: 8=2*2*2=2^3

, so we can rewrite our first term as (2x)^3

. Since our second term does not have a exact cubic root, we must rewrite as $\sqrt[3]{243} ^{3}$ . Now we have a=2x

and $b= \sqrt[3]{243}$ , so lets use the sum of cube identity to rewrite our expression:
a^3+b^3=(a+b)(a^2-ab+b^2)

$(2x)^3+ \sqrt[3]{243} ^3=(2x+ \sqrt[3]{243} )((2x)^2-2x \sqrt[3]{243} + \sqrt[3]{243} ^2$
$(2x)^3+ \sqrt[3]{243} ^3=(2x+ 3\sqrt[3]{3^2} )((4x^2-6x \sqrt[3]{3^2} +27 \sqrt[3]{3} )$
$(2x)^3+ \sqrt[3]{243} ^3=(2x+ 3\sqrt[3]{9} )((4x^2-6x \sqrt[3]{9} +27 \sqrt[3]{3} )$

We can conclude that we can use the sum of cubes identity to rewrite the expression 8x^3+243 as $(2x+ 3\sqrt[3]{9} )((4x^2-6x \sqrt[3]{9} +27 \sqrt[3]{3} )$

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What identity can be used to rewrite the expression 8x^3+243

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