Basic Math Examples

$3 (10^{6 y}) = 11 (10^{3 y}) + 4$

Step 1

Subtract $11 \cdot 10^{3 y}$ from both sides of the equation.

$3 \cdot 10^{6 y} - 11 \cdot 10^{3 y} = 4$

Step 2

Rewrite $10^{6 y}$ as exponentiation.

$3 \cdot {(10^{y})}^{6} - 11 \cdot 10^{3 y} = 4$

Step 3

Rewrite $10^{3 y}$ as exponentiation.

$3 \cdot {(10^{y})}^{6} - 11 \cdot {(10^{y})}^{3} = 4$

Step 4

Substitute $u$ for $10^{y}$ .

$3 \cdot u^{6} - 11 \cdot u^{3} = 4$

Step 5

Multiply $- 11$ by $u^{3}$ .

$3 u^{6} - 11 u^{3} = 4$

Step 6

Solve for $u$ .

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Step 6.1

Subtract $4$ from both sides of the equation.

$3 u^{6} - 11 u^{3} - 4 = 0$

Step 6.2

Factor the left side of the equation.

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Step 6.2.1

Rewrite $u^{6}$ as ${(u^{3})}^{2}$ .

$3 {(u^{3})}^{2} - 11 u^{3} - 4 = 0$

Step 6.2.2

Let $u = u^{3}$ . Substitute $u$ for all occurrences of $u^{3}$ .

$3 u^{2} - 11 u - 4 = 0$

Step 6.2.3

Factor by grouping.

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Step 6.2.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = - 11$ .

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Step 6.2.3.1.1

Factor $- 11$ out of $- 11 u$ .

$3 u^{2} - 11 u - 4 = 0$

Step 6.2.3.1.2

Rewrite $- 11$ as $1$ plus $- 12$

$3 u^{2} + (1 - 12) u - 4 = 0$

Step 6.2.3.1.3

Apply the distributive property.

$3 u^{2} + 1 u - 12 u - 4 = 0$

Step 6.2.3.1.4

Multiply $u$ by $1$ .

$3 u^{2} + u - 12 u - 4 = 0$

Step 6.2.3.2

Factor out the greatest common factor from each group.

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Step 6.2.3.2.1

Group the first two terms and the last two terms.

$(3 u^{2} + u) - 12 u - 4 = 0$

Step 6.2.3.2.2

Factor out the greatest common factor (GCF) from each group.

$u (3 u + 1) - 4 (3 u + 1) = 0$

Step 6.2.3.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$(3 u + 1) (u - 4) = 0$

Step 6.2.4

Replace all occurrences of $u$ with $u^{3}$ .

$(3 u^{3} + 1) (u^{3} - 4) = 0$

Step 6.3

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

$3 u^{3} + 1 = 0$

$u^{3} - 4 = 0$

Step 6.4

Set $3 u^{3} + 1$ equal to $0$ and solve for $u$ .

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Step 6.4.1

Set $3 u^{3} + 1$ equal to $0$ .

$3 u^{3} + 1 = 0$

Step 6.4.2

Solve $3 u^{3} + 1 = 0$ for $u$ .

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Step 6.4.2.1

Subtract $1$ from both sides of the equation.

$3 u^{3} = - 1$

Step 6.4.2.2

Divide each term in $3 u^{3} = - 1$ by $3$ and simplify.

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Step 6.4.2.2.1

Divide each term in $3 u^{3} = - 1$ by $3$ .

$\frac{3 u^{3}}{3} = \frac{- 1}{3}$

Step 6.4.2.2.2

Simplify the left side.

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Step 6.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 6.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 u^{3}}{3} = \frac{- 1}{3}$

Step 6.4.2.2.2.1.2

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

$u^{3} = \frac{- 1}{3}$

Step 6.4.2.2.3

Simplify the right side.

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Step 6.4.2.2.3.1

Move the negative in front of the fraction.

$u^{3} = - \frac{1}{3}$

Step 6.4.2.3

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

$u = \sqrt[3]{- \frac{1}{3}}$

Step 6.4.2.4

Simplify $\sqrt[3]{- \frac{1}{3}}$ .

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Step 6.4.2.4.1

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

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Step 6.4.2.4.1.1

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

$u = \sqrt[3]{{(- 1)}^{3} \frac{1}{3}}$

Step 6.4.2.4.1.2

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

$u = \sqrt[3]{{({(- 1)}^{3})}^{3} \frac{1}{3}}$

Step 6.4.2.4.2

Pull terms out from under the radical.

$u = {(- 1)}^{3} \sqrt[3]{\frac{1}{3}}$

Step 6.4.2.4.3

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

$u = - \sqrt[3]{\frac{1}{3}}$

Step 6.4.2.4.4

Rewrite $\sqrt[3]{\frac{1}{3}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{3}}$ .

$u = - \frac{\sqrt[3]{1}}{\sqrt[3]{3}}$

Step 6.4.2.4.5

Any root of $1$ is $1$ .

$u = - \frac{1}{\sqrt[3]{3}}$

Step 6.4.2.4.6

Multiply $\frac{1}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

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

Step 6.4.2.4.7

Combine and simplify the denominator.

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Step 6.4.2.4.7.1

Multiply $\frac{1}{\sqrt[3]{3}}$ by $\frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{2}}$ .

$u = - \frac{{\sqrt[3]{3}}^{2}}{\sqrt[3]{3} {\sqrt[3]{3}}^{2}}$

Step 6.4.2.4.7.2

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

$u = - \frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1} {\sqrt[3]{3}}^{2}}$

Step 6.4.2.4.7.3

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

$u = - \frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{1 + 2}}$

Step 6.4.2.4.7.4

Add $1$ and $2$ .

$u = - \frac{{\sqrt[3]{3}}^{2}}{{\sqrt[3]{3}}^{3}}$

Step 6.4.2.4.7.5

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

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Step 6.4.2.4.7.5.1

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

$u = - \frac{{\sqrt[3]{3}}^{2}}{{(3^{\frac{1}{3}})}^{3}}$

Step 6.4.2.4.7.5.2

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

$u = - \frac{{\sqrt[3]{3}}^{2}}{3^{\frac{1}{3} \cdot 3}}$

Step 6.4.2.4.7.5.3

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

$u = - \frac{{\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}}$

Step 6.4.2.4.7.5.4

Cancel the common factor of $3$ .

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Step 6.4.2.4.7.5.4.1

Cancel the common factor.

$u = - \frac{{\sqrt[3]{3}}^{2}}{3^{\frac{3}{3}}}$

Step 6.4.2.4.7.5.4.2

Rewrite the expression.

$u = - \frac{{\sqrt[3]{3}}^{2}}{3^{1}}$

Step 6.4.2.4.7.5.5

Evaluate the exponent.

$u = - \frac{{\sqrt[3]{3}}^{2}}{3}$

Step 6.4.2.4.8

Simplify the numerator.

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Step 6.4.2.4.8.1

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

$u = - \frac{\sqrt[3]{3^{2}}}{3}$

Step 6.4.2.4.8.2

Raise $3$ to the power of $2$ .

$u = - \frac{\sqrt[3]{9}}{3}$

Step 6.5

Set $u^{3} - 4$ equal to $0$ and solve for $u$ .

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Step 6.5.1

Set $u^{3} - 4$ equal to $0$ .

$u^{3} - 4 = 0$

Step 6.5.2

Solve $u^{3} - 4 = 0$ for $u$ .

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Step 6.5.2.1

Add $4$ to both sides of the equation.

$u^{3} = 4$

Step 6.5.2.2

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

$u = \sqrt[3]{4}$

Step 6.6

The final solution is all the values that make $(3 u^{3} + 1) (u^{3} - 4) = 0$ true.

$u = - \frac{\sqrt[3]{9}}{3}, \sqrt[3]{4}$

Step 7

Substitute $- \frac{\sqrt[3]{9}}{3}$ for $u$ in $u = 10^{y}$ .

$- \frac{\sqrt[3]{9}}{3} = 10^{y}$

Step 8

Solve $- \frac{\sqrt[3]{9}}{3} = 10^{y}$ .

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Step 8.1

Rewrite the equation as $10^{y} = - \frac{\sqrt[3]{9}}{3}$ .

$10^{y} = - \frac{\sqrt[3]{9}}{3}$

Step 8.2

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{y}) = \log (- \frac{\sqrt[3]{9}}{3})$

Step 8.3

The equation cannot be solved because $\log (- \frac{\sqrt[3]{9}}{3})$ is undefined.

Undefined

Step 8.4

There is no solution for $10^{y} = - \frac{\sqrt[3]{9}}{3}$

No solution

Step 9

Substitute $\sqrt[3]{4}$ for $u$ in $u = 10^{y}$ .

$\sqrt[3]{4} = 10^{y}$

Step 10

Solve $\sqrt[3]{4} = 10^{y}$ .

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Step 10.1

Rewrite the equation as $10^{y} = \sqrt[3]{4}$ .

$10^{y} = \sqrt[3]{4}$

Step 10.2

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{y}) = \log (\sqrt[3]{4})$

Step 10.3

Expand the left side.

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Step 10.3.1

Expand $\log (10^{y})$ by moving $y$ outside the logarithm.

$y \log (10) = \log (\sqrt[3]{4})$

Step 10.3.2

Logarithm base $10$ of $10$ is $1$ .

$y \cdot 1 = \log (\sqrt[3]{4})$

Step 10.3.3

Multiply $y$ by $1$ .

$y = \log (\sqrt[3]{4})$

Step 11

The result can be shown in multiple forms.

Exact Form:

$y = \log (\sqrt[3]{4})$

Decimal Form:

$y = 0.20068666 \dots$