Calculus Examples

$30 = 6 {(x^{2} + 27000)}^{\frac{1}{3}} - 200$

Step 1

Rewrite the equation as $6 {(x^{2} + 27000)}^{\frac{1}{3}} - 200 = 30$ .

$6 {(x^{2} + 27000)}^{\frac{1}{3}} - 200 = 30$

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Add $200$ to both sides of the equation.

$6 {(x^{2} + 27000)}^{\frac{1}{3}} = 30 + 200$

Step 2.2

Add $30$ and $200$ .

$6 {(x^{2} + 27000)}^{\frac{1}{3}} = 230$

Step 3

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${(6 {(x^{2} + 27000)}^{\frac{1}{3}})}^{3} = 230^{3}$

Step 4

Simplify the exponent.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify ${(6 {(x^{2} + 27000)}^{\frac{1}{3}})}^{3}$ .

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Step 4.1.1.1

Apply the product rule to $6 {(x^{2} + 27000)}^{\frac{1}{3}}$ .

$6^{3} {({(x^{2} + 27000)}^{\frac{1}{3}})}^{3} = 230^{3}$

Step 4.1.1.2

Raise $6$ to the power of $3$ .

$216 {({(x^{2} + 27000)}^{\frac{1}{3}})}^{3} = 230^{3}$

Step 4.1.1.3

Multiply the exponents in ${({(x^{2} + 27000)}^{\frac{1}{3}})}^{3}$ .

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Step 4.1.1.3.1

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

$216 {(x^{2} + 27000)}^{\frac{1}{3} \cdot 3} = 230^{3}$

Step 4.1.1.3.2

Cancel the common factor of $3$ .

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Step 4.1.1.3.2.1

Cancel the common factor.

$216 {(x^{2} + 27000)}^{\frac{1}{3} \cdot 3} = 230^{3}$

Step 4.1.1.3.2.2

Rewrite the expression.

$216 {(x^{2} + 27000)}^{1} = 230^{3}$

Step 4.1.1.4

Simplify.

$216 (x^{2} + 27000) = 230^{3}$

Step 4.1.1.5

Apply the distributive property.

$216 x^{2} + 216 \cdot 27000 = 230^{3}$

Step 4.1.1.6

Multiply $216$ by $27000$ .

$216 x^{2} + 5832000 = 230^{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Raise $230$ to the power of $3$ .

$216 x^{2} + 5832000 = 12167000$

Step 5

Solve for $x$ .

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Step 5.1

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1.1

Subtract $5832000$ from both sides of the equation.

$216 x^{2} = 12167000 - 5832000$

Step 5.1.2

Subtract $5832000$ from $12167000$ .

$216 x^{2} = 6335000$

Step 5.2

Divide each term in $216 x^{2} = 6335000$ by $216$ and simplify.

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Step 5.2.1

Divide each term in $216 x^{2} = 6335000$ by $216$ .

$\frac{216 x^{2}}{216} = \frac{6335000}{216}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $216$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{216 x^{2}}{216} = \frac{6335000}{216}$

Step 5.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{6335000}{216}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Cancel the common factor of $6335000$ and $216$ .

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Step 5.2.3.1.1

Factor $8$ out of $6335000$ .

$x^{2} = \frac{8 (791875)}{216}$

Step 5.2.3.1.2

Cancel the common factors.

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Step 5.2.3.1.2.1

Factor $8$ out of $216$ .

$x^{2} = \frac{8 \cdot 791875}{8 \cdot 27}$

Step 5.2.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{8 \cdot 791875}{8 \cdot 27}$

Step 5.2.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{791875}{27}$

Step 5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{791875}{27}}$

Step 5.4

Simplify $\pm \sqrt{\frac{791875}{27}}$ .

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Step 5.4.1

Rewrite $\sqrt{\frac{791875}{27}}$ as $\frac{\sqrt{791875}}{\sqrt{27}}$ .

$x = \pm \frac{\sqrt{791875}}{\sqrt{27}}$

Step 5.4.2

Simplify the numerator.

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Step 5.4.2.1

Rewrite $791875$ as $25^{2} \cdot 1267$ .

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Step 5.4.2.1.1

Factor $625$ out of $791875$ .

$x = \pm \frac{\sqrt{625 (1267)}}{\sqrt{27}}$

Step 5.4.2.1.2

Rewrite $625$ as $25^{2}$ .

$x = \pm \frac{\sqrt{25^{2} \cdot 1267}}{\sqrt{27}}$

Step 5.4.2.2

Pull terms out from under the radical.

$x = \pm \frac{25 \sqrt{1267}}{\sqrt{27}}$

Step 5.4.3

Simplify the denominator.

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Step 5.4.3.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 5.4.3.1.1

Factor $9$ out of $27$ .

$x = \pm \frac{25 \sqrt{1267}}{\sqrt{9 (3)}}$

Step 5.4.3.1.2

Rewrite $9$ as $3^{2}$ .

$x = \pm \frac{25 \sqrt{1267}}{\sqrt{3^{2} \cdot 3}}$

Step 5.4.3.2

Pull terms out from under the radical.

$x = \pm \frac{25 \sqrt{1267}}{3 \sqrt{3}}$

Step 5.4.4

Multiply $\frac{25 \sqrt{1267}}{3 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{25 \sqrt{1267}}{3 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.4.5

Combine and simplify the denominator.

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Step 5.4.5.1

Multiply $\frac{25 \sqrt{1267}}{3 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 \sqrt{3} \sqrt{3}}$

Step 5.4.5.2

Move $\sqrt{3}$ .

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 (\sqrt{3} \sqrt{3})}$

Step 5.4.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 5.4.5.4

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 5.4.5.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 {\sqrt{3}}^{1 + 1}}$

Step 5.4.5.6

Add $1$ and $1$ .

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 {\sqrt{3}}^{2}}$

Step 5.4.5.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.4.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 {(3^{\frac{1}{2}})}^{2}}$

Step 5.4.5.7.2

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

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 5.4.5.7.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 \cdot 3^{\frac{2}{2}}}$

Step 5.4.5.7.4

Cancel the common factor of $2$ .

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Step 5.4.5.7.4.1

Cancel the common factor.

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 \cdot 3^{\frac{2}{2}}}$

Step 5.4.5.7.4.2

Rewrite the expression.

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 \cdot 3^{1}}$

Step 5.4.5.7.5

Evaluate the exponent.

$x = \pm \frac{25 \sqrt{1267} \sqrt{3}}{3 \cdot 3}$

Step 5.4.6

Simplify the numerator.

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Step 5.4.6.1

Combine using the product rule for radicals.

$x = \pm \frac{25 \sqrt{3 \cdot 1267}}{3 \cdot 3}$

Step 5.4.6.2

Multiply $3$ by $1267$ .

$x = \pm \frac{25 \sqrt{3801}}{3 \cdot 3}$

Step 5.4.7

Multiply $3$ by $3$ .

$x = \pm \frac{25 \sqrt{3801}}{9}$

Step 5.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{25 \sqrt{3801}}{9}$

Step 5.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{25 \sqrt{3801}}{9}$

Step 5.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{25 \sqrt{3801}}{9}, - \frac{25 \sqrt{3801}}{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{25 \sqrt{3801}}{9}, - \frac{25 \sqrt{3801}}{9}$

Decimal Form:

$x = 171.25625157 \dots, - 171.25625157 \dots$