Calculus Examples

$45 = 6 {(x^{2} + 30000)}^{\frac{1}{3}} - 160$

Step 1

Rewrite the equation as $6 {(x^{2} + 30000)}^{\frac{1}{3}} - 160 = 45$ .

$6 {(x^{2} + 30000)}^{\frac{1}{3}} - 160 = 45$

Step 2

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

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Step 2.1

Add $160$ to both sides of the equation.

$6 {(x^{2} + 30000)}^{\frac{1}{3}} = 45 + 160$

Step 2.2

Add $45$ and $160$ .

$6 {(x^{2} + 30000)}^{\frac{1}{3}} = 205$

Step 3

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

${(6 {(x^{2} + 30000)}^{\frac{1}{3}})}^{3} = 205^{3}$

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 ${(6 {(x^{2} + 30000)}^{\frac{1}{3}})}^{3}$ .

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Step 4.1.1.1

Apply the product rule to $6 {(x^{2} + 30000)}^{\frac{1}{3}}$ .

$6^{3} {({(x^{2} + 30000)}^{\frac{1}{3}})}^{3} = 205^{3}$

Step 4.1.1.2

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

$216 {({(x^{2} + 30000)}^{\frac{1}{3}})}^{3} = 205^{3}$

Step 4.1.1.3

Multiply the exponents in ${({(x^{2} + 30000)}^{\frac{1}{3}})}^{3}$ .

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Step 4.1.1.3.1

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

$216 {(x^{2} + 30000)}^{\frac{1}{3} \cdot 3} = 205^{3}$

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.

$216 {(x^{2} + 30000)}^{\frac{1}{3} \cdot 3} = 205^{3}$

Step 4.1.1.3.2.2

Rewrite the expression.

$216 {(x^{2} + 30000)}^{1} = 205^{3}$

Step 4.1.1.4

Simplify.

$216 (x^{2} + 30000) = 205^{3}$

Step 4.1.1.5

Apply the distributive property.

$216 x^{2} + 216 \cdot 30000 = 205^{3}$

Step 4.1.1.6

Multiply $216$ by $30000$ .

$216 x^{2} + 6480000 = 205^{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Raise $205$ to the power of $3$ .

$216 x^{2} + 6480000 = 8615125$

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 $6480000$ from both sides of the equation.

$216 x^{2} = 8615125 - 6480000$

Step 5.1.2

Subtract $6480000$ from $8615125$ .

$216 x^{2} = 2135125$

Step 5.2

Divide each term in $216 x^{2} = 2135125$ by $216$ and simplify.

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Step 5.2.1

Divide each term in $216 x^{2} = 2135125$ by $216$ .

$\frac{216 x^{2}}{216} = \frac{2135125}{216}$

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.

$\frac{216 x^{2}}{216} = \frac{2135125}{216}$

Step 5.2.2.1.2

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

$x^{2} = \frac{2135125}{216}$

Step 5.3

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

$x = \pm \sqrt{\frac{2135125}{216}}$

Step 5.4

Simplify $\pm \sqrt{\frac{2135125}{216}}$ .

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Step 5.4.1

Rewrite $\sqrt{\frac{2135125}{216}}$ as $\frac{\sqrt{2135125}}{\sqrt{216}}$ .

$x = \pm \frac{\sqrt{2135125}}{\sqrt{216}}$

Step 5.4.2

Simplify the numerator.

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Step 5.4.2.1

Rewrite $2135125$ as $5^{2} \cdot 85405$ .

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Step 5.4.2.1.1

Factor $25$ out of $2135125$ .

$x = \pm \frac{\sqrt{25 (85405)}}{\sqrt{216}}$

Step 5.4.2.1.2

Rewrite $25$ as $5^{2}$ .

$x = \pm \frac{\sqrt{5^{2} \cdot 85405}}{\sqrt{216}}$

Step 5.4.2.2

Pull terms out from under the radical.

$x = \pm \frac{5 \sqrt{85405}}{\sqrt{216}}$

Step 5.4.3

Simplify the denominator.

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Step 5.4.3.1

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 5.4.3.1.1

Factor $36$ out of $216$ .

$x = \pm \frac{5 \sqrt{85405}}{\sqrt{36 (6)}}$

Step 5.4.3.1.2

Rewrite $36$ as $6^{2}$ .

$x = \pm \frac{5 \sqrt{85405}}{\sqrt{6^{2} \cdot 6}}$

Step 5.4.3.2

Pull terms out from under the radical.

$x = \pm \frac{5 \sqrt{85405}}{6 \sqrt{6}}$

Step 5.4.4

Multiply $\frac{5 \sqrt{85405}}{6 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{5 \sqrt{85405}}{6 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 5.4.5

Combine and simplify the denominator.

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Step 5.4.5.1

Multiply $\frac{5 \sqrt{85405}}{6 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 \sqrt{6} \sqrt{6}}$

Step 5.4.5.2

Move $\sqrt{6}$ .

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 (\sqrt{6} \sqrt{6})}$

Step 5.4.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 ({\sqrt{6}}^{1} \sqrt{6})}$

Step 5.4.5.4

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}$

Step 5.4.5.5

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

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 {\sqrt{6}}^{1 + 1}}$

Step 5.4.5.6

Add $1$ and $1$ .

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 {\sqrt{6}}^{2}}$

Step 5.4.5.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 5.4.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 {(6^{\frac{1}{2}})}^{2}}$

Step 5.4.5.7.2

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

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 5.4.5.7.3

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

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 \cdot 6^{\frac{2}{2}}}$

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.

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 \cdot 6^{\frac{2}{2}}}$

Step 5.4.5.7.4.2

Rewrite the expression.

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 \cdot 6^{1}}$

Step 5.4.5.7.5

Evaluate the exponent.

$x = \pm \frac{5 \sqrt{85405} \sqrt{6}}{6 \cdot 6}$

Step 5.4.6

Simplify the numerator.

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Step 5.4.6.1

Combine using the product rule for radicals.

$x = \pm \frac{5 \sqrt{6 \cdot 85405}}{6 \cdot 6}$

Step 5.4.6.2

Multiply $6$ by $85405$ .

$x = \pm \frac{5 \sqrt{512430}}{6 \cdot 6}$

Step 5.4.7

Multiply $6$ by $6$ .

$x = \pm \frac{5 \sqrt{512430}}{36}$

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.

$x = \frac{5 \sqrt{512430}}{36}$

Step 5.5.2

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

$x = - \frac{5 \sqrt{512430}}{36}$

Step 5.5.3

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

$x = \frac{5 \sqrt{512430}}{36}, - \frac{5 \sqrt{512430}}{36}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \sqrt{512430}}{36}, - \frac{5 \sqrt{512430}}{36}$

Decimal Form:

$x = 99.42252241 \dots, - 99.42252241 \dots$