Question

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

Answer 1

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

Answer 2

Answer: Sum of Cubes

Step-by-step explanation:

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