Algebra Examples

$\sqrt[3]{8 x^{3} - 1} + 1 = 2 x$

Step 1

Simplify each term.

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Step 1.1

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

$\sqrt[3]{{(2 x)}^{3} - 1} + 1 = 2 x$

Step 1.2

Rewrite $1$ as $1^{3}$ .

$\sqrt[3]{{(2 x)}^{3} - 1^{3}} + 1 = 2 x$

Step 1.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 2 x$ and $b = 1$ .

$\sqrt[3]{(2 x - 1) ({(2 x)}^{2} + 2 x \cdot 1 + 1^{2})} + 1 = 2 x$

Step 1.4

Simplify.

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Step 1.4.1

Apply the product rule to $2 x$ .

$\sqrt[3]{(2 x - 1) (2^{2} x^{2} + 2 x \cdot 1 + 1^{2})} + 1 = 2 x$

Step 1.4.2

Raise $2$ to the power of $2$ .

$\sqrt[3]{(2 x - 1) (4 x^{2} + 2 x \cdot 1 + 1^{2})} + 1 = 2 x$

Step 1.4.3

Multiply $2$ by $1$ .

$\sqrt[3]{(2 x - 1) (4 x^{2} + 2 x + 1^{2})} + 1 = 2 x$

Step 1.4.4

One to any power is one.

$\sqrt[3]{(2 x - 1) (4 x^{2} + 2 x + 1)} + 1 = 2 x$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 0$

Step 3