Linear Algebra Examples

$x = 8 y^{2}$

Step 1

Rewrite the equation as $8 y^{2} = x$ .

$8 y^{2} = x$

Step 2

Divide each term in $8 y^{2} = x$ by $8$ and simplify.

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Step 2.1

Divide each term in $8 y^{2} = x$ by $8$ .

$\frac{8 y^{2}}{8} = \frac{x}{8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $8$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{8 y^{2}}{8} = \frac{x}{8}$

Step 2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{x}{8}$

Step 3

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

$y = \pm \sqrt{\frac{x}{8}}$

Step 4

Simplify $\pm \sqrt{\frac{x}{8}}$ .

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Step 4.1

Rewrite $\frac{x}{8}$ as ${(\frac{1}{2})}^{2} \frac{x}{2}$ .

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Step 4.1.1

Factor the perfect power $1^{2}$ out of $x$ .

$y = \pm \sqrt{\frac{1^{2} x}{8}}$

Step 4.1.2

Factor the perfect power $2^{2}$ out of $8$ .

$y = \pm \sqrt{\frac{1^{2} x}{2^{2} \cdot 2}}$

Step 4.1.3

Rearrange the fraction $\frac{1^{2} x}{2^{2} \cdot 2}$ .

$y = \pm \sqrt{{(\frac{1}{2})}^{2} \frac{x}{2}}$

Step 4.2

Pull terms out from under the radical.

$y = \pm \frac{1}{2} \sqrt{\frac{x}{2}}$

Step 4.3

Rewrite $\sqrt{\frac{x}{2}}$ as $\frac{\sqrt{x}}{\sqrt{2}}$ .

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

Step 4.4

Multiply $\frac{\sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{1}{2} (\frac{\sqrt{x}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{\sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 4.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 4.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 4.5.5

Add $1$ and $1$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.5.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{x} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 4.5.6.2

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

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{x} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 4.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{x} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.5.6.4

Cancel the common factor of $2$ .

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Step 4.5.6.4.1

Cancel the common factor.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{x} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.5.6.4.2

Rewrite the expression.

$y = \pm \frac{1}{2} \cdot \frac{\sqrt{x} \sqrt{2}}{2^{1}}$

Step 4.5.6.5

Evaluate the exponent.

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

Step 4.6

Combine using the product rule for radicals.

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

Step 4.7

Multiply $\frac{1}{2} \cdot \frac{\sqrt{x \cdot 2}}{2}$ .

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Step 4.7.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{x \cdot 2}}{2}$ .

$y = \pm \frac{\sqrt{x \cdot 2}}{2 \cdot 2}$

Step 4.7.2

Multiply $2$ by $2$ .

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

Step 4.8

Reorder factors in $\pm \frac{\sqrt{x \cdot 2}}{4}$ .

$y = \pm \frac{\sqrt{2 x}}{4}$

Step 5

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

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Step 5.1

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

$y = \frac{\sqrt{2 x}}{4}$

Step 5.2

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

$y = - \frac{\sqrt{2 x}}{4}$

Step 5.3

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

$y = \frac{\sqrt{2 x}}{4}$

$y = - \frac{\sqrt{2 x}}{4}$

$y = \frac{\sqrt{2 x}}{4}$

$y = - \frac{\sqrt{2 x}}{4}$

Step 6

Set the radicand in $\sqrt{2 x}$ greater than or equal to $0$ to find where the expression is defined.

$2 x \geq 0$

Step 7

Divide each term in $2 x \geq 0$ by $2$ and simplify.

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Step 7.1

Divide each term in $2 x \geq 0$ by $2$ .

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $2$ .

$x \geq 0$

Step 8

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 9