Algebra Examples

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

Step 1

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

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

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

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

Step 2.2

Divide $6$ by $2$ .

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

Step 2.3

Simplify the left side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

Multiply $3$ by $2$ .

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

Step 2.4

Simplify the right side.

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Step 2.4.1

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

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Step 2.4.1.1

Apply the product rule to $- y^{3}$ .

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

Step 2.4.1.2

Raise $- 1$ to the power of $2$ .

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

Step 2.4.1.3

Multiply ${(y^{3})}^{2}$ by $1$ .

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

Step 2.4.1.4

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

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Step 2.4.1.4.1

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

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

Step 2.4.1.4.2

Multiply $3$ by $2$ .

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

Step 3

Solve for $y$ .

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Step 3.1

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

$| y | = | y |$

Step 3.2

Solve for $y$ .

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Step 3.2.1

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

$y = y$

$y = - y$

$- y = y$

$- y = - y$

Step 3.2.2

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

$y = y$

$y = - y$

Step 3.2.3

Solve $y = y$ for $y$ .

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Step 3.2.3.1

Move all terms containing $y$ to the left side of the equation.

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Step 3.2.3.1.1

Subtract $y$ from both sides of the equation.

$y - y = 0$

Step 3.2.3.1.2

Subtract $y$ from $y$ .

$0 = 0$

Step 3.2.3.2

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

Always true

Step 3.2.4

Solve $y = - y$ for $y$ .

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Step 3.2.4.1

Move all terms containing $y$ to the left side of the equation.

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Step 3.2.4.1.1

Add $y$ to both sides of the equation.

$y + y = 0$

Step 3.2.4.1.2

Add $y$ and $y$ .

$2 y = 0$

Step 3.2.4.2

Divide each term in $2 y = 0$ by $2$ and simplify.

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Step 3.2.4.2.1

Divide each term in $2 y = 0$ by $2$ .

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

Step 3.2.4.2.2

Simplify the left side.

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Step 3.2.4.2.2.1

Cancel the common factor of $2$ .

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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.

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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.

$y = 0$