Algebra Examples

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

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

Step 1

Divide each term in

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

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

5^{3}

$5^{3}$ and simplify.

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Step 1.1

Divide each term in

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

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

5^{3}

$5^{3}$ .

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of

5^{3}

$5^{3}$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Divide

$t^{3}$ by

$1$ .

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Raise

$50$ to the power of

$3$ .

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

Step 1.3.2

Raise

$5$ to the power of

$3$ .

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

Step 1.3.3

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.

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Step 3.1

Rewrite

$1000$ as

$10^{3}$ .

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

Step 3.2

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

$b = 10$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Move

$10$ to the left of

$t$ .

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

Step 3.3.2

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

$t$ .

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Step 5.1

Set

$t - 10$ equal to

$0$ .

$t - 10 = 0$

Step 5.2

Add

$10$ to both sides of the equation.

$t = 10$

Step 6

Set

$t^{2} + 10 t + 100$ equal to

$0$ and solve for

$t$ .

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Step 6.1

Set

$t^{2} + 10 t + 100$ equal to

$0$ .

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

Step 6.2

Solve

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

$t$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

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

Step 6.2.2

Substitute the values

$a = 1$ ,

$b = 10$ , and

$c = 100$ into the quadratic formula and solve for

$t$ .

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

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Raise

$10$ to the power of

$2$ .

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

Step 6.2.3.1.2

Multiply

$- 4 \cdot 1 \cdot 100$ .

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Step 6.2.3.1.2.1

Multiply

$- 4$ by

$1$ .

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

Step 6.2.3.1.2.2

Multiply

$- 4$ by

$100$ .

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

Step 6.2.3.1.3

Subtract

$400$ from

$100$ .

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

Step 6.2.3.1.4

Rewrite

$- 300$ as

$- 1 (300)$ .

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

Step 6.2.3.1.5

Rewrite

$\sqrt{- 1 (300)}$ as

$\sqrt{- 1} \cdot \sqrt{300}$ .

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

Step 6.2.3.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

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

Step 6.2.3.1.7

Rewrite

$300$ as

$10^{2} \cdot 3$ .

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Step 6.2.3.1.7.1

Factor

$100$ out of

$300$ .

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

Step 6.2.3.1.7.2

Rewrite

$100$ as

$10^{2}$ .

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

Step 6.2.3.1.8

Pull terms out from under the radical.

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

Step 6.2.3.1.9

Move

$10$ to the left of

$i$ .

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

Step 6.2.3.2

Multiply

$2$ by

$1$ .

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

Step 6.2.3.3

Simplify

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

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

Step 6.2.4

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.

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