Basic Math Examples

$\sqrt[3]{2 t^{2} - 8 t} = - 2$

Step 1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{2 t^{2} - 8 t}}^{3} = {(- 2)}^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 t^{2} - 8 t}$ as ${(2 t^{2} - 8 t)}^{\frac{1}{3}}$ .

${({(2 t^{2} - 8 t)}^{\frac{1}{3}})}^{3} = {(- 2)}^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(2 t^{2} - 8 t)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(2 t^{2} - 8 t)}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

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

${(2 t^{2} - 8 t)}^{\frac{1}{3} \cdot 3} = {(- 2)}^{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(2 t^{2} - 8 t)}^{\frac{1}{3} \cdot 3} = {(- 2)}^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 t^{2} - 8 t)}^{1} = {(- 2)}^{3}$

Step 2.2.1.2

Simplify.

$2 t^{2} - 8 t = {(- 2)}^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$2 t^{2} - 8 t = - 8$

Step 3

Solve for $t$ .

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Step 3.1

Add $8$ to both sides of the equation.

$2 t^{2} - 8 t + 8 = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $2$ out of $2 t^{2} - 8 t + 8$ .

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Step 3.2.1.1

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

$2 (t^{2}) - 8 t + 8 = 0$

Step 3.2.1.2

Factor $2$ out of $- 8 t$ .

$2 (t^{2}) + 2 (- 4 t) + 8 = 0$

Step 3.2.1.3

Factor $2$ out of $8$ .

$2 t^{2} + 2 (- 4 t) + 2 \cdot 4 = 0$

Step 3.2.1.4

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

$2 (t^{2} - 4 t) + 2 \cdot 4 = 0$

Step 3.2.1.5

Factor $2$ out of $2 (t^{2} - 4 t) + 2 \cdot 4$ .

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

Step 3.2.2

Factor using the perfect square rule.

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Step 3.2.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 t = 2 \cdot t \cdot 2$

Step 3.2.2.3

Rewrite the polynomial.

$2 (t^{2} - 2 \cdot t \cdot 2 + 2^{2}) = 0$

Step 3.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = t$ and $b = 2$ .

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

Step 3.3

Divide each term in $2 {(t - 2)}^{2} = 0$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 {(t - 2)}^{2} = 0$ by $2$ .

$\frac{2 {(t - 2)}^{2}}{2} = \frac{0}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 {(t - 2)}^{2}}{2} = \frac{0}{2}$

Step 3.3.2.1.2

Divide ${(t - 2)}^{2}$ by $1$ .

${(t - 2)}^{2} = \frac{0}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $2$ .

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

Step 3.4

Set the $t - 2$ equal to $0$ .

$t - 2 = 0$

Step 3.5

Add $2$ to both sides of the equation.

$t = 2$