Algebra Examples

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

Step 1

Rewrite the equation as $\frac{1}{\sqrt{x}} = y$ .

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

Step 2

Cross multiply.

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Step 2.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Multiply $y$ by $\sqrt{x}$ .

$y \sqrt{x} = 1$

Step 3

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

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

Step 4

Simplify each side of the equation.

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(y x^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the product rule to $y x^{\frac{1}{2}}$ .

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

Step 4.2.1.2

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.2.1

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

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

Step 4.2.1.2.2

Cancel the common factor of $2$ .

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Step 4.2.1.2.2.1

Cancel the common factor.

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

Step 4.2.1.2.2.2

Rewrite the expression.

$y^{2} x^{1} = 1^{2}$

Step 4.2.1.3

Simplify.

$y^{2} x = 1^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

One to any power is one.

$y^{2} x = 1$

Step 5

Divide each term in $y^{2} x = 1$ by $y^{2}$ and simplify.

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Step 5.1

Divide each term in $y^{2} x = 1$ by $y^{2}$ .

$\frac{y^{2} x}{y^{2}} = \frac{1}{y^{2}}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

Cancel the common factor.

$\frac{y^{2} x}{y^{2}} = \frac{1}{y^{2}}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{y^{2}}$

Step 6

To rewrite as a function of $y$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $y$ is on the other side.

$f (y) = \frac{1}{y^{2}}$