Algebra Examples

$y = \frac{y {(y^{3})}^{2}}{y^{3}}$

Step 1

Factor each term.

Tap for more steps...

Step 1.1

Reduce the expression $\frac{y {(y^{3})}^{2}}{y^{3}}$ by cancelling the common factors.

Tap for more steps...

Step 1.1.1

Factor $y$ out of $y {(y^{3})}^{2}$ .

$y = \frac{y ({(y^{3})}^{2})}{y^{3}}$

Step 1.1.2

Factor $y$ out of $y^{3}$ .

$y = \frac{y ({(y^{3})}^{2})}{y \cdot y^{2}}$

Step 1.1.3

Cancel the common factor.

$y = \frac{y {(y^{3})}^{2}}{y \cdot y^{2}}$

Step 1.1.4

Rewrite the expression.

$y = \frac{{(y^{3})}^{2}}{y^{2}}$

Step 1.2

Multiply the exponents in ${(y^{3})}^{2}$ .

Tap for more steps...

Step 1.2.1

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

$y = \frac{y^{3 \cdot 2}}{y^{2}}$

Step 1.2.2

Multiply $3$ by $2$ .

$y = \frac{y^{6}}{y^{2}}$

Step 1.3

Cancel the common factor of $y^{6}$ and $y^{2}$ .

Tap for more steps...

Step 1.3.1

Factor $y^{2}$ out of $y^{6}$ .

$y = \frac{y^{2} y^{4}}{y^{2}}$

Step 1.3.2

Cancel the common factors.

Tap for more steps...

Step 1.3.2.1

Multiply by $1$ .

$y = \frac{y^{2} y^{4}}{y^{2} \cdot 1}$

Step 1.3.2.2

Cancel the common factor.

$y = \frac{y^{2} y^{4}}{y^{2} \cdot 1}$

Step 1.3.2.3

Rewrite the expression.

$y = \frac{y^{4}}{1}$

Step 1.3.2.4

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

$y = y^{4}$

Step 2

Solve the equation.

Tap for more steps...

Step 2.1

Subtract $y^{4}$ from both sides of the equation.

$y - y^{4} = 0$

Step 2.2

Factor the left side of the equation.

Tap for more steps...

Step 2.2.1

Factor $y$ out of $y - y^{4}$ .

Tap for more steps...

Step 2.2.1.1

Raise $y$ to the power of $1$ .

$y - y^{4} = 0$

Step 2.2.1.2

Factor $y$ out of $y^{1}$ .

$y \cdot 1 - y^{4} = 0$

Step 2.2.1.3

Factor $y$ out of $- y^{4}$ .

$y \cdot 1 + y (- y^{3}) = 0$

Step 2.2.1.4

Factor $y$ out of $y \cdot 1 + y (- y^{3})$ .

$y (1 - y^{3}) = 0$

Step 2.2.2

Rewrite $1$ as $1^{3}$ .

$y (1^{3} - y^{3}) = 0$

Step 2.2.3

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 $a = 1$ and $b = y$ .

$y ((1 - y) (1^{2} + 1 y + y^{2})) = 0$

Step 2.2.4

Factor.

Tap for more steps...

Step 2.2.4.1

Simplify.

Tap for more steps...

Step 2.2.4.1.1

One to any power is one.

$y ((1 - y) (1 + 1 y + y^{2})) = 0$

Step 2.2.4.1.2

Multiply $y$ by $1$ .

$y ((1 - y) (1 + y + y^{2})) = 0$

Step 2.2.4.2

Remove unnecessary parentheses.

$y (1 - y) (1 + y + y^{2}) = 0$

Step 2.3

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

$y = 0$

$1 - y = 0$

$1 + y + y^{2} = 0$

Step 2.4

Set $y$ equal to $0$ .

$y = 0$

Step 2.5

Set $1 - y$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 2.5.1

Set $1 - y$ equal to $0$ .

$1 - y = 0$

Step 2.5.2

Solve $1 - y = 0$ for $y$ .

Tap for more steps...

Step 2.5.2.1

Subtract $1$ from both sides of the equation.

$- y = - 1$

Step 2.5.2.2

Divide each term in $- y = - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 2.5.2.2.1

Divide each term in $- y = - 1$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 1}{- 1}$

Step 2.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 1}{- 1}$

Step 2.5.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 1}{- 1}$

Step 2.5.2.2.3

Simplify the right side.

Tap for more steps...

Step 2.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$y = 1$

Step 2.6

Set $1 + y + y^{2}$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 2.6.1

Set $1 + y + y^{2}$ equal to $0$ .

$1 + y + y^{2} = 0$

Step 2.6.2

Solve $1 + y + y^{2} = 0$ for $y$ .

Tap for more steps...

Step 2.6.2.1

Use the quadratic formula to find the solutions.

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

Step 2.6.2.2

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

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

Step 2.6.2.3

Simplify.

Tap for more steps...

Step 2.6.2.3.1

Simplify the numerator.

Tap for more steps...

Step 2.6.2.3.1.1

One to any power is one.

$y = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.6.2.3.1.2

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

Tap for more steps...

Step 2.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.6.2.3.1.2.2

Multiply $- 4$ by $1$ .

$y = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.6.2.3.1.3

Subtract $4$ from $1$ .

$y = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.6.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.6.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$y = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.6.2.3.1.6

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

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

Step 2.6.2.3.2

Multiply $2$ by $1$ .

$y = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 2.6.2.4

The final answer is the combination of both solutions.

$y = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 2.7

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

$y = 0, 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$