Solve for t 5^3*t^3=50^3

$5^{3} \cdot t^{3} = 50^{3}$

Step 1

Divide each term in $(5^{3} \cdot t^{3}) = 50^{3}$ by $5^{3}$ and simplify.

Divide each term in $(5^{3} \cdot t^{3}) = 50^{3}$ by $5^{3}$ .

$\frac{(5^{3} \cdot t^{3})}{5^{3}} = \frac{50^{3}}{5^{3}}$

Simplify the left side.

Cancel the common factor of $5^{3}$ .

Cancel the common factor.

$\frac{5^{3} \cdot t^{3}}{5^{3}} = \frac{50^{3}}{5^{3}}$

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

$t^{3} = \frac{50^{3}}{5^{3}}$

Simplify the right side.

Raise $50$ to the power of $3$ .

$t^{3} = \frac{125000}{5^{3}}$

Raise $5$ to the power of $3$ .

$t^{3} = \frac{125000}{125}$

Divide $125000$ by $125$ .

$t^{3} = 1000$

Step 2

Subtract $1000$ from both sides of the equation.

$t^{3} - 1000 = 0$

Step 3

Factor the left side of the equation.

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

$t^{3} - 10^{3} = 0$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = t$ and $b = 10$ .

$(t - 10) (t^{2} + t \cdot 10 + 10^{2}) = 0$

Simplify.

Move $10$ to the left of $t$ .

$(t - 10) (t^{2} + 10 t + 10^{2}) = 0$

Raise $10$ to the power of $2$ .

$(t - 10) (t^{2} + 10 t + 100) = 0$

Step 4

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

$t - 10 = 0$

$t^{2} + 10 t + 100 = 0$

Step 5

Set $t - 10$ equal to $0$ and solve for $t$ .

Set $t - 10$ equal to $0$ .

$t - 10 = 0$

Add $10$ to both sides of the equation.

$t = 10$

Step 6

Set $t^{2} + 10 t + 100$ equal to $0$ and solve for $t$ .

Set $t^{2} + 10 t + 100$ equal to $0$ .

$t^{2} + 10 t + 100 = 0$

Solve $t^{2} + 10 t + 100 = 0$ for $t$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 10$ , and $c = 100$ into the quadratic formula and solve for $t$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (1 \cdot 100)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $10$ to the power of $2$ .

$t = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot 100}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 100$ .

Multiply $- 4$ by $1$ .

$t = \frac{- 10 \pm \sqrt{100 - 4 \cdot 100}}{2 \cdot 1}$

Multiply $- 4$ by $100$ .

$t = \frac{- 10 \pm \sqrt{100 - 400}}{2 \cdot 1}$

Subtract $400$ from $100$ .

$t = \frac{- 10 \pm \sqrt{- 300}}{2 \cdot 1}$

Rewrite $- 300$ as $- 1 (300)$ .

$t = \frac{- 10 \pm \sqrt{- 1 \cdot 300}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (300)}$ as $\sqrt{- 1} \cdot \sqrt{300}$ .

$t = \frac{- 10 \pm \sqrt{- 1} \cdot \sqrt{300}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$t = \frac{- 10 \pm i \cdot \sqrt{300}}{2 \cdot 1}$

Rewrite $300$ as $10^{2} \cdot 3$ .

Factor $100$ out of $300$ .

$t = \frac{- 10 \pm i \cdot \sqrt{100 (3)}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$t = \frac{- 10 \pm i \cdot \sqrt{10^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$t = \frac{- 10 \pm i \cdot (10 \sqrt{3})}{2 \cdot 1}$

Move $10$ to the left of $i$ .

$t = \frac{- 10 \pm 10 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$t = \frac{- 10 \pm 10 i \sqrt{3}}{2}$

Simplify $\frac{- 10 \pm 10 i \sqrt{3}}{2}$ .

$t = - 5 \pm 5 i \sqrt{3}$

The final answer is the combination of both solutions.

$t = - 5 + 5 i \sqrt{3}, - 5 - 5 i \sqrt{3}$

Step 7

The final solution is all the values that make $(t - 10) (t^{2} + 10 t + 100) = 0$ true.

$t = 10, - 5 + 5 i \sqrt{3}, - 5 - 5 i \sqrt{3}$

Solve for t 5^3*t^3=50^3

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