Basic Math Examples

$6 t^{2 + 5} = 2 t^{2 + 1}$

Step 1

Simplify $6 t^{2 + 5}$ .

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Step 1.1

Rewrite.

$0 + 0 + 6 t^{2 + 5} = 2 t^{2 + 1}$

Step 1.2

Simplify by adding zeros.

$6 t^{2 + 5} = 2 t^{2 + 1}$

Step 1.3

Add $2$ and $5$ .

$6 t^{7} = 2 t^{2 + 1}$

Step 2

Add $2$ and $1$ .

$6 t^{7} = 2 t^{3}$

Step 3

Subtract $2 t^{3}$ from both sides of the equation.

$6 t^{7} - 2 t^{3} = 0$

Step 4

Factor $2 t^{3}$ out of $6 t^{7} - 2 t^{3}$ .

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Step 4.1

Factor $2 t^{3}$ out of $6 t^{7}$ .

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

Step 4.2

Factor $2 t^{3}$ out of $- 2 t^{3}$ .

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

Step 4.3

Factor $2 t^{3}$ out of $2 t^{3} (3 t^{4}) + 2 t^{3} (- 1)$ .

$2 t^{3} (3 t^{4} - 1) = 0$

Step 5

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

$t^{3} = 0$

$3 t^{4} - 1 = 0$

Step 6

Set $t^{3}$ equal to $0$ and solve for $t$ .

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Step 6.1

Set $t^{3}$ equal to $0$ .

$t^{3} = 0$

Step 6.2

Solve $t^{3} = 0$ for $t$ .

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Step 6.2.1

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

$t = \sqrt[3]{0}$

Step 6.2.2

Simplify $\sqrt[3]{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{3}$ .

$t = \sqrt[3]{0^{3}}$

Step 6.2.2.2

Pull terms out from under the radical, assuming real numbers.

$t = 0$

Step 7

Set $3 t^{4} - 1$ equal to $0$ and solve for $t$ .

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Step 7.1

Set $3 t^{4} - 1$ equal to $0$ .

$3 t^{4} - 1 = 0$

Step 7.2

Solve $3 t^{4} - 1 = 0$ for $t$ .

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Step 7.2.1

Add $1$ to both sides of the equation.

$3 t^{4} = 1$

Step 7.2.2

Divide each term in $3 t^{4} = 1$ by $3$ and simplify.

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Step 7.2.2.1

Divide each term in $3 t^{4} = 1$ by $3$ .

$\frac{3 t^{4}}{3} = \frac{1}{3}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{3 t^{4}}{3} = \frac{1}{3}$

Step 7.2.2.2.1.2

Divide $t^{4}$ by $1$ .

$t^{4} = \frac{1}{3}$

Step 7.2.3

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

$t = \pm \sqrt[4]{\frac{1}{3}}$

Step 7.2.4

Simplify $\pm \sqrt[4]{\frac{1}{3}}$ .

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Step 7.2.4.1

Rewrite $\sqrt[4]{\frac{1}{3}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{3}}$ .

$t = \pm \frac{\sqrt[4]{1}}{\sqrt[4]{3}}$

Step 7.2.4.2

Any root of $1$ is $1$ .

$t = \pm \frac{1}{\sqrt[4]{3}}$

Step 7.2.4.3

Multiply $\frac{1}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$t = \pm \frac{1}{\sqrt[4]{3}} \cdot \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$

Step 7.2.4.4

Combine and simplify the denominator.

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Step 7.2.4.4.1

Multiply $\frac{1}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$t = \pm \frac{{\sqrt[4]{3}}^{3}}{\sqrt[4]{3} {\sqrt[4]{3}}^{3}}$

Step 7.2.4.4.2

Raise $\sqrt[4]{3}$ to the power of $1$ .

$t = \pm \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1} {\sqrt[4]{3}}^{3}}$

Step 7.2.4.4.3

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

$t = \pm \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1 + 3}}$

Step 7.2.4.4.4

Add $1$ and $3$ .

$t = \pm \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{4}}$

Step 7.2.4.4.5

Rewrite ${\sqrt[4]{3}}^{4}$ as $3$ .

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Step 7.2.4.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

$t = \pm \frac{{\sqrt[4]{3}}^{3}}{{(3^{\frac{1}{4}})}^{4}}$

Step 7.2.4.4.5.2

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

$t = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{\frac{1}{4} \cdot 4}}$

Step 7.2.4.4.5.3

Combine $\frac{1}{4}$ and $4$ .

$t = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 7.2.4.4.5.4

Cancel the common factor of $4$ .

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Step 7.2.4.4.5.4.1

Cancel the common factor.

$t = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 7.2.4.4.5.4.2

Rewrite the expression.

$t = \pm \frac{{\sqrt[4]{3}}^{3}}{3^{1}}$

Step 7.2.4.4.5.5

Evaluate the exponent.

$t = \pm \frac{{\sqrt[4]{3}}^{3}}{3}$

Step 7.2.4.5

Simplify the numerator.

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Step 7.2.4.5.1

Rewrite ${\sqrt[4]{3}}^{3}$ as $\sqrt[4]{3^{3}}$ .

$t = \pm \frac{\sqrt[4]{3^{3}}}{3}$

Step 7.2.4.5.2

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

$t = \pm \frac{\sqrt[4]{27}}{3}$

Step 7.2.5

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

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Step 7.2.5.1

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

$t = \frac{\sqrt[4]{27}}{3}$

Step 7.2.5.2

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

$t = - \frac{\sqrt[4]{27}}{3}$

Step 7.2.5.3

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

$t = \frac{\sqrt[4]{27}}{3}, - \frac{\sqrt[4]{27}}{3}$

Step 8

The final solution is all the values that make $2 t^{3} (3 t^{4} - 1) = 0$ true.

$t = 0, \frac{\sqrt[4]{27}}{3}, - \frac{\sqrt[4]{27}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$t = 0, \frac{\sqrt[4]{27}}{3}, - \frac{\sqrt[4]{27}}{3}$

Decimal Form:

$t = 0, 0.75983568 \dots, - 0.75983568 \dots$