Linear Algebra Examples

$0 = 0.0241 x^{2} x + 5.5$

Step 1

Rewrite the equation as $0.0241 x^{2} x + 5.5 = 0$ .

$0.0241 x^{2} x + 5.5 = 0$

Step 2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1

Move $x$ .

$0.0241 (x \cdot x^{2}) + 5.5 = 0$

Step 2.2

Multiply $x$ by $x^{2}$ .

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Step 2.2.1

Raise $x$ to the power of $1$ .

$0.0241 (x^{1} x^{2}) + 5.5 = 0$

Step 2.2.2

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

$0.0241 x^{1 + 2} + 5.5 = 0$

Step 2.3

Add $1$ and $2$ .

$0.0241 x^{3} + 5.5 = 0$

Step 3

Subtract $5.5$ from both sides of the equation.

$0.0241 x^{3} = - 5.5$

Step 4

Add $5.5$ to both sides of the equation.

$0.0241 x^{3} + 5.5 = 0$

Step 5

Factor $0.0001$ out of $0.0241 x^{3} + 5.5$ .

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Step 5.1

Factor $0.0001$ out of $0.0241 x^{3}$ .

$0.0001 (241 x^{3}) + 5.5 = 0$

Step 5.2

Factor $0.0001$ out of $5.5$ .

$0.0001 (241 x^{3}) + 0.0001 (55000) = 0$

Step 5.3

Factor $0.0001$ out of $0.0001 (241 x^{3}) + 0.0001 (55000)$ .

$0.0001 (241 x^{3} + 55000) = 0$

Step 6

Divide each term in $0.0001 (241 x^{3} + 55000) = 0$ by $0.0001$ and simplify.

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Step 6.1

Divide each term in $0.0001 (241 x^{3} + 55000) = 0$ by $0.0001$ .

$\frac{0.0001 (241 x^{3} + 55000)}{0.0001} = \frac{0}{0.0001}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $0.0001$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{0.0001 (241 x^{3} + 55000)}{0.0001} = \frac{0}{0.0001}$

Step 6.2.1.2

Divide $241 x^{3} + 55000$ by $1$ .

$241 x^{3} + 55000 = \frac{0}{0.0001}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0$ by $0.0001$ .

$241 x^{3} + 55000 = 0$

Step 7

Subtract $55000$ from both sides of the equation.

$241 x^{3} = - 55000$

Step 8

Divide each term in $241 x^{3} = - 55000$ by $241$ and simplify.

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Step 8.1

Divide each term in $241 x^{3} = - 55000$ by $241$ .

$\frac{241 x^{3}}{241} = \frac{- 55000}{241}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $241$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{241 x^{3}}{241} = \frac{- 55000}{241}$

Step 8.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{- 55000}{241}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

$x^{3} = - \frac{55000}{241}$

Step 9

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

$x = \sqrt[3]{- \frac{55000}{241}}$

Step 10

Simplify $\sqrt[3]{- \frac{55000}{241}}$ .

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Step 10.1

Rewrite $- \frac{55000}{241}$ as ${({(- 1)}^{3})}^{3} \frac{55000}{241}$ .

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Step 10.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \frac{55000}{241}}$

Step 10.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{({(- 1)}^{3})}^{3} \frac{55000}{241}}$

Step 10.2

Pull terms out from under the radical.

$x = {(- 1)}^{3} \sqrt[3]{\frac{55000}{241}}$

Step 10.3

Raise $- 1$ to the power of $3$ .

$x = - \sqrt[3]{\frac{55000}{241}}$

Step 10.4

Rewrite $\sqrt[3]{\frac{55000}{241}}$ as $\frac{\sqrt[3]{55000}}{\sqrt[3]{241}}$ .

$x = - \frac{\sqrt[3]{55000}}{\sqrt[3]{241}}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Rewrite $55000$ as $10^{3} \cdot 55$ .

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Step 10.5.1.1

Factor $1000$ out of $55000$ .

$x = - \frac{\sqrt[3]{1000 (55)}}{\sqrt[3]{241}}$

Step 10.5.1.2

Rewrite $1000$ as $10^{3}$ .

$x = - \frac{\sqrt[3]{10^{3} \cdot 55}}{\sqrt[3]{241}}$

Step 10.5.2

Pull terms out from under the radical.

$x = - \frac{10 \sqrt[3]{55}}{\sqrt[3]{241}}$

Step 10.6

Multiply $\frac{10 \sqrt[3]{55}}{\sqrt[3]{241}}$ by $\frac{{\sqrt[3]{241}}^{2}}{{\sqrt[3]{241}}^{2}}$ .

$x = - (\frac{10 \sqrt[3]{55}}{\sqrt[3]{241}} \cdot \frac{{\sqrt[3]{241}}^{2}}{{\sqrt[3]{241}}^{2}})$

Step 10.7

Combine and simplify the denominator.

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Step 10.7.1

Multiply $\frac{10 \sqrt[3]{55}}{\sqrt[3]{241}}$ by $\frac{{\sqrt[3]{241}}^{2}}{{\sqrt[3]{241}}^{2}}$ .

$x = - \frac{10 \sqrt[3]{55} {\sqrt[3]{241}}^{2}}{\sqrt[3]{241} {\sqrt[3]{241}}^{2}}$

Step 10.7.2

Raise $\sqrt[3]{241}$ to the power of $1$ .

$x = - \frac{10 \sqrt[3]{55} {\sqrt[3]{241}}^{2}}{{\sqrt[3]{241}}^{1} {\sqrt[3]{241}}^{2}}$

Step 10.7.3

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

$x = - \frac{10 \sqrt[3]{55} {\sqrt[3]{241}}^{2}}{{\sqrt[3]{241}}^{1 + 2}}$

Step 10.7.4

Add $1$ and $2$ .

$x = - \frac{10 \sqrt[3]{55} {\sqrt[3]{241}}^{2}}{{\sqrt[3]{241}}^{3}}$

Step 10.7.5

Rewrite ${\sqrt[3]{241}}^{3}$ as $241$ .

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Step 10.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{241}$ as $241^{\frac{1}{3}}$ .

$x = - \frac{10 \sqrt[3]{55} {\sqrt[3]{241}}^{2}}{{(241^{\frac{1}{3}})}^{3}}$

Step 10.7.5.2

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

$x = - \frac{10 \sqrt[3]{55} {\sqrt[3]{241}}^{2}}{241^{\frac{1}{3} \cdot 3}}$

Step 10.7.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = - \frac{10 \sqrt[3]{55} {\sqrt[3]{241}}^{2}}{241^{\frac{3}{3}}}$

Step 10.7.5.4

Cancel the common factor of $3$ .

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Step 10.7.5.4.1

Cancel the common factor.

$x = - \frac{10 \sqrt[3]{55} {\sqrt[3]{241}}^{2}}{241^{\frac{3}{3}}}$

Step 10.7.5.4.2

Rewrite the expression.

$x = - \frac{10 \sqrt[3]{55} {\sqrt[3]{241}}^{2}}{241^{1}}$

Step 10.7.5.5

Evaluate the exponent.

$x = - \frac{10 \sqrt[3]{55} {\sqrt[3]{241}}^{2}}{241}$

Step 10.8

Simplify the numerator.

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Step 10.8.1

Rewrite ${\sqrt[3]{241}}^{2}$ as $\sqrt[3]{241^{2}}$ .

$x = - \frac{10 \sqrt[3]{55} \sqrt[3]{241^{2}}}{241}$

Step 10.8.2

Raise $241$ to the power of $2$ .

$x = - \frac{10 \sqrt[3]{55} \sqrt[3]{58081}}{241}$

Step 10.8.3

Combine exponents.

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Step 10.8.3.1

Combine using the product rule for radicals.

$x = - \frac{10 \sqrt[3]{58081 \cdot 55}}{241}$

Step 10.8.3.2

Multiply $58081$ by $55$ .

$x = - \frac{10 \sqrt[3]{3194455}}{241}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{10 \sqrt[3]{3194455}}{241}$

Decimal Form:

$x = - 6.11104125 \dots$