Algebra Examples

\sqrt{y^{6}} = - y^{3}

$\sqrt{y^{6}} = - y^{3}$

Step 1

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

{\sqrt{y^{6}}}^{2} = {(- y^{3})}^{2}

${\sqrt{y^{6}}}^{2} = {(- y^{3})}^{2}$

Step 2

Simplify each side of the equation.

Tap for more steps...

Step 2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{y^{6}}

$\sqrt{y^{6}}$ as

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

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

{(y^{\frac{6}{2}})}^{2} = {(- y^{3})}^{2}

${(y^{\frac{6}{2}})}^{2} = {(- y^{3})}^{2}$

Step 2.2

Divide

6

$6$ by

2

$2$ .

{(y^{3})}^{2} = {(- y^{3})}^{2}

${(y^{3})}^{2} = {(- y^{3})}^{2}$

Step 2.3

Simplify the left side.

Tap for more steps...

Step 2.3.1

Multiply the exponents in

{(y^{3})}^{2}

${(y^{3})}^{2}$ .

Tap for more steps...

Step 2.3.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

y^{3 \cdot 2} = {(- y^{3})}^{2}

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

Step 2.3.1.2

Multiply

3

$3$ by

2

$2$ .

y^{6} = {(- y^{3})}^{2}

$y^{6} = {(- y^{3})}^{2}$

y^{6} = {(- y^{3})}^{2}

$y^{6} = {(- y^{3})}^{2}$

y^{6} = {(- y^{3})}^{2}

$y^{6} = {(- y^{3})}^{2}$

Step 2.4

Simplify the right side.

Tap for more steps...

Step 2.4.1

Simplify

{(- y^{3})}^{2}

${(- y^{3})}^{2}$ .

Tap for more steps...

Step 2.4.1.1

Apply the product rule to

- y^{3}

$- y^{3}$ .

y^{6} = {(- 1)}^{2} {(y^{3})}^{2}

$y^{6} = {(- 1)}^{2} {(y^{3})}^{2}$

Step 2.4.1.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

y^{6} = 1 {(y^{3})}^{2}

$y^{6} = 1 {(y^{3})}^{2}$

Step 2.4.1.3

Multiply

{(y^{3})}^{2}

${(y^{3})}^{2}$ by

1

$1$ .

y^{6} = {(y^{3})}^{2}

$y^{6} = {(y^{3})}^{2}$

Step 2.4.1.4

Multiply the exponents in

{(y^{3})}^{2}

${(y^{3})}^{2}$ .

Tap for more steps...

Step 2.4.1.4.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

y^{6} = y^{3 \cdot 2}

$y^{6} = y^{3 \cdot 2}$

Step 2.4.1.4.2

Multiply

3

$3$ by

2

$2$ .

y^{6} = y^{6}

$y^{6} = y^{6}$

y^{6} = y^{6}

$y^{6} = y^{6}$

y^{6} = y^{6}

$y^{6} = y^{6}$

y^{6} = y^{6}

$y^{6} = y^{6}$

y^{6} = y^{6}

$y^{6} = y^{6}$

Step 3

Solve for

y

$y$ .

Tap for more steps...

Step 3.1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

| y | = | y |

$| y | = | y |$

Step 3.2

Solve for

y

$y$ .

Tap for more steps...

Step 3.2.1

Rewrite the absolute value equation as four equations without absolute value bars.

y = y

$y = y$

y = - y

$y = - y$

- y = y

$- y = y$

- y = - y

$- y = - y$

Step 3.2.2

After simplifying, there are only two unique equations to be solved.

y = y

$y = y$

y = - y

$y = - y$

Step 3.2.3

Solve

y = y

$y = y$ for

y

$y$ .

Tap for more steps...

Step 3.2.3.1

Move all terms containing

y

$y$ to the left side of the equation.

Tap for more steps...

Step 3.2.3.1.1

Subtract

y

$y$ from both sides of the equation.

y - y = 0

$y - y = 0$

Step 3.2.3.1.2

Subtract

y

$y$ from

y

$y$ .

0 = 0

$0 = 0$

0 = 0

$0 = 0$

Step 3.2.3.2

Since

0 = 0

$0 = 0$ , the equation will always be true.

Always true

Step 3.2.4

Solve

y = - y

$y = - y$ for

y

$y$ .

Tap for more steps...

Step 3.2.4.1

Move all terms containing

y

$y$ to the left side of the equation.

Tap for more steps...

Step 3.2.4.1.1

Add

y

$y$ to both sides of the equation.

y + y = 0

$y + y = 0$

Step 3.2.4.1.2

Add

y

$y$ and

y

$y$ .

2 y = 0

$2 y = 0$

2 y = 0

$2 y = 0$

Step 3.2.4.2

Divide each term in

2 y = 0

$2 y = 0$ by

2

$2$ and simplify.

Tap for more steps...

Step 3.2.4.2.1

Divide each term in

2 y = 0

$2 y = 0$ by

2

$2$ .

\frac{2 y}{2} = \frac{0}{2}

$\frac{2 y}{2} = \frac{0}{2}$

Step 3.2.4.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.4.2.2.1

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 3.2.4.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{0}{2}$

Step 3.2.4.2.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{0}{2}$

Step 3.2.4.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.4.2.3.1

Divide

$0$ by

$2$ .

$y = 0$

Step 3.2.5

List all of the solutions.

$y = 0$

Step 4

Verify each of the solutions by substituting them into

$\sqrt{y^{6}} = - y^{3}$ and solving.

$y = 0$