Basic Math Examples

$\sqrt[4]{4 y^{2} - 3} = - y$

Step 1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{4 y^{2} - 3}}^{4} = {(- y)}^{4}$

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 $\sqrt[4]{4 y^{2} - 3}$ as ${(4 y^{2} - 3)}^{\frac{1}{4}}$ .

${({(4 y^{2} - 3)}^{\frac{1}{4}})}^{4} = {(- y)}^{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(4 y^{2} - 3)}^{\frac{1}{4}})}^{4}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(4 y^{2} - 3)}^{\frac{1}{4}})}^{4}$ .

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Step 2.2.1.1.1

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

${(4 y^{2} - 3)}^{\frac{1}{4} \cdot 4} = {(- y)}^{4}$

Step 2.2.1.1.2

Cancel the common factor of $4$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(4 y^{2} - 3)}^{\frac{1}{4} \cdot 4} = {(- y)}^{4}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(4 y^{2} - 3)}^{1} = {(- y)}^{4}$

Step 2.2.1.2

Simplify.

$4 y^{2} - 3 = {(- y)}^{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(- y)}^{4}$ .

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Step 2.3.1.1

Apply the product rule to $- y$ .

$4 y^{2} - 3 = {(- 1)}^{4} y^{4}$

Step 2.3.1.2

Raise $- 1$ to the power of $4$ .

$4 y^{2} - 3 = 1 y^{4}$

Step 2.3.1.3

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

$4 y^{2} - 3 = y^{4}$

Step 3

Solve for $y$ .

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Step 3.1

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

$4 y^{2} - 3 - y^{4} = 0$

Step 3.2

Substitute $u = y^{2}$ into the equation. This will make the quadratic formula easy to use.

$- u^{2} + 4 u - 3 = 0$

$u = y^{2}$

Step 3.3

Factor the left side of the equation.

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Step 3.3.1

Factor $- 1$ out of $- u^{2} + 4 u - 3$ .

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Step 3.3.1.1

Factor $- 1$ out of $- u^{2}$ .

$- (u^{2}) + 4 u - 3 = 0$

Step 3.3.1.2

Factor $- 1$ out of $4 u$ .

$- (u^{2}) - (- 4 u) - 3 = 0$

Step 3.3.1.3

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

$- (u^{2}) - (- 4 u) - 1 \cdot 3 = 0$

Step 3.3.1.4

Factor $- 1$ out of $- (u^{2}) - (- 4 u)$ .

$- (u^{2} - 4 u) - 1 \cdot 3 = 0$

Step 3.3.1.5

Factor $- 1$ out of $- (u^{2} - 4 u) - 1 (3)$ .

$- (u^{2} - 4 u + 3) = 0$

Step 3.3.2

Factor.

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Step 3.3.2.1

Factor $u^{2} - 4 u + 3$ using the AC method.

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Step 3.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 3.3.2.1.2

Write the factored form using these integers.

$- ((u - 3) (u - 1)) = 0$

Step 3.3.2.2

Remove unnecessary parentheses.

$- (u - 3) (u - 1) = 0$

Step 3.4

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

$u - 3 = 0$

$u - 1 = 0$

Step 3.5

Set $u - 3$ equal to $0$ and solve for $u$ .

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Step 3.5.1

Set $u - 3$ equal to $0$ .

$u - 3 = 0$

Step 3.5.2

Add $3$ to both sides of the equation.

$u = 3$

Step 3.6

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

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Step 3.6.1

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

$u - 1 = 0$

Step 3.6.2

Add $1$ to both sides of the equation.

$u = 1$

Step 3.7

The final solution is all the values that make $- (u - 3) (u - 1) = 0$ true.

$u = 3, 1$

Step 3.8

Substitute the real value of $u = y^{2}$ back into the solved equation.

$y^{2} = 3$

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

Step 3.9

Solve the first equation for $y$ .

$y^{2} = 3$

Step 3.10

Solve the equation for $y$ .

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Step 3.10.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{3}$

Step 3.10.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.10.2.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{3}$

Step 3.10.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{3}$

Step 3.10.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{3}, - \sqrt{3}$

Step 3.11

Solve the second equation for $y$ .

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

Step 3.12

Solve the equation for $y$ .

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Step 3.12.1

Remove parentheses.

$y^{2} = 1$

Step 3.12.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{1}$

Step 3.12.3

Any root of $1$ is $1$ .

$y = \pm 1$

Step 3.12.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.12.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y = 1$

Step 3.12.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - 1$

Step 3.12.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = 1, - 1$

Step 3.13

The solution to $- y^{4} + 4 y^{2} - 3 = 0$ is $y = \sqrt{3}, - \sqrt{3}, 1, - 1$ .

$y = \sqrt{3}, - \sqrt{3}, 1, - 1$

Step 4

Exclude the solutions that do not make $\sqrt[4]{4 y^{2} - 3} = - y$ true.

$y = - \sqrt{3}, - 1$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = - \sqrt{3}, - 1$

Decimal Form:

$y = - 1.73205080 \dots, - 1$