Calculus Examples

$12288 \sqrt{x} = 192 x^{2}$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${(12288 \sqrt{x})}^{2} = {(192 x^{2})}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(12288 x^{\frac{1}{2}})}^{2} = {(192 x^{2})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(12288 x^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply the product rule to $12288 x^{\frac{1}{2}}$ .

$12288^{2} {(x^{\frac{1}{2}})}^{2} = {(192 x^{2})}^{2}$

Step 2.2.1.2

Raise $12288$ to the power of $2$ .

$150994944 {(x^{\frac{1}{2}})}^{2} = {(192 x^{2})}^{2}$

Step 2.2.1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$150994944 x^{\frac{1}{2} \cdot 2} = {(192 x^{2})}^{2}$

Step 2.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.1.3.2.1

Cancel the common factor.

$150994944 x^{\frac{1}{2} \cdot 2} = {(192 x^{2})}^{2}$

Step 2.2.1.3.2.2

Rewrite the expression.

$150994944 x^{1} = {(192 x^{2})}^{2}$

Step 2.2.1.4

Simplify.

$150994944 x = {(192 x^{2})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(192 x^{2})}^{2}$ .

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Step 2.3.1.1

Apply the product rule to $192 x^{2}$ .

$150994944 x = 192^{2} {(x^{2})}^{2}$

Step 2.3.1.2

Raise $192$ to the power of $2$ .

$150994944 x = 36864 {(x^{2})}^{2}$

Step 2.3.1.3

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.3.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$150994944 x = 36864 x^{2 \cdot 2}$

Step 2.3.1.3.2

Multiply $2$ by $2$ .

$150994944 x = 36864 x^{4}$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $36864 x^{4}$ from both sides of the equation.

$150994944 x - 36864 x^{4} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $36864 x$ out of $150994944 x - 36864 x^{4}$ .

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Step 3.2.1.1

Factor $36864 x$ out of $150994944 x$ .

$36864 x (4096) - 36864 x^{4} = 0$

Step 3.2.1.2

Factor $36864 x$ out of $- 36864 x^{4}$ .

$36864 x (4096) + 36864 x (- x^{3}) = 0$

Step 3.2.1.3

Factor $36864 x$ out of $36864 x (4096) + 36864 x (- x^{3})$ .

$36864 x (4096 - x^{3}) = 0$

Step 3.2.2

Rewrite $4096$ as $16^{3}$ .

$36864 x (16^{3} - x^{3}) = 0$

Step 3.2.3

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 $a = 16$ and $b = x$ .

$36864 x ((16 - x) (16^{2} + 16 x + x^{2})) = 0$

Step 3.2.4

Factor.

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Step 3.2.4.1

Raise $16$ to the power of $2$ .

$36864 x ((16 - x) (256 + 16 x + x^{2})) = 0$

Step 3.2.4.2

Remove unnecessary parentheses.

$36864 x (16 - x) (256 + 16 x + x^{2}) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$16 - x = 0$

$256 + 16 x + x^{2} = 0$

Step 3.4

Set $x$ equal to $0$ .

$x = 0$

Step 3.5

Set $16 - x$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $16 - x$ equal to $0$ .

$16 - x = 0$

Step 3.5.2

Solve $16 - x = 0$ for $x$ .

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Step 3.5.2.1

Subtract $16$ from both sides of the equation.

$- x = - 16$

Step 3.5.2.2

Divide each term in $- x = - 16$ by $- 1$ and simplify.

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Step 3.5.2.2.1

Divide each term in $- x = - 16$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 16}{- 1}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 16}{- 1}$

Step 3.5.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 16}{- 1}$

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Divide $- 16$ by $- 1$ .

$x = 16$

Step 3.6

Set $256 + 16 x + x^{2}$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $256 + 16 x + x^{2}$ equal to $0$ .

$256 + 16 x + x^{2} = 0$

Step 3.6.2

Solve $256 + 16 x + x^{2} = 0$ for $x$ .

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Step 3.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.6.2.2

Substitute the values $a = 1$ , $b = 16$ , and $c = 256$ into the quadratic formula and solve for $x$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (1 \cdot 256)}}{2 \cdot 1}$

Step 3.6.2.3

Simplify.

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Step 3.6.2.3.1

Simplify the numerator.

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Step 3.6.2.3.1.1

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 1 \cdot 256}}{2 \cdot 1}$

Step 3.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 256$ .

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Step 3.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 256}}{2 \cdot 1}$

Step 3.6.2.3.1.2.2

Multiply $- 4$ by $256$ .

$x = \frac{- 16 \pm \sqrt{256 - 1024}}{2 \cdot 1}$

Step 3.6.2.3.1.3

Subtract $1024$ from $256$ .

$x = \frac{- 16 \pm \sqrt{- 768}}{2 \cdot 1}$

Step 3.6.2.3.1.4

Rewrite $- 768$ as $- 1 (768)$ .

$x = \frac{- 16 \pm \sqrt{- 1 \cdot 768}}{2 \cdot 1}$

Step 3.6.2.3.1.5

Rewrite $\sqrt{- 1 (768)}$ as $\sqrt{- 1} \cdot \sqrt{768}$ .

$x = \frac{- 16 \pm \sqrt{- 1} \cdot \sqrt{768}}{2 \cdot 1}$

Step 3.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 16 \pm i \cdot \sqrt{768}}{2 \cdot 1}$

Step 3.6.2.3.1.7

Rewrite $768$ as $16^{2} \cdot 3$ .

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Step 3.6.2.3.1.7.1

Factor $256$ out of $768$ .

$x = \frac{- 16 \pm i \cdot \sqrt{256 (3)}}{2 \cdot 1}$

Step 3.6.2.3.1.7.2

Rewrite $256$ as $16^{2}$ .

$x = \frac{- 16 \pm i \cdot \sqrt{16^{2} \cdot 3}}{2 \cdot 1}$

Step 3.6.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 16 \pm i \cdot (16 \sqrt{3})}{2 \cdot 1}$

Step 3.6.2.3.1.9

Move $16$ to the left of $i$ .

$x = \frac{- 16 \pm 16 i \sqrt{3}}{2 \cdot 1}$

Step 3.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 16 \pm 16 i \sqrt{3}}{2}$

Step 3.6.2.3.3

Simplify $\frac{- 16 \pm 16 i \sqrt{3}}{2}$ .

$x = - 8 \pm 8 i \sqrt{3}$

Step 3.6.2.4

The final answer is the combination of both solutions.

$x = - 8 + 8 i \sqrt{3}, - 8 - 8 i \sqrt{3}$

Step 3.7

The final solution is all the values that make $36864 x (16 - x) (256 + 16 x + x^{2}) = 0$ true.

$x = 0, 16, - 8 + 8 i \sqrt{3}, - 8 - 8 i \sqrt{3}$