Basic Math Examples

$2 y \sqrt{\frac{1}{4}} - \frac{1}{y^{2}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$2 y \frac{\sqrt{1}}{\sqrt{4}} - \frac{1}{y^{2}}$

Step 1.2

Any root of $1$ is $1$ .

$2 y \frac{1}{\sqrt{4}} - \frac{1}{y^{2}}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $4$ as $2^{2}$ .

$2 y \frac{1}{\sqrt{2^{2}}} - \frac{1}{y^{2}}$

Step 1.3.2

Pull terms out from under the radical, assuming positive real numbers.

$2 y \frac{1}{2} - \frac{1}{y^{2}}$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Factor $2$ out of $2 y$ .

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

Step 1.4.2

Cancel the common factor.

$2 y \frac{1}{2} - \frac{1}{y^{2}}$

Step 1.4.3

Rewrite the expression.

$y - \frac{1}{y^{2}}$

Step 2

To write $y$ as a fraction with a common denominator, multiply by $\frac{y^{2}}{y^{2}}$ .

$\frac{y \cdot y^{2}}{y^{2}} - \frac{1}{y^{2}}$

Step 3

Combine the numerators over the common denominator.

$\frac{y \cdot y^{2} - 1}{y^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 4.1.1

Multiply $y$ by $y^{2}$ .

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Step 4.1.1.1

Raise $y$ to the power of $1$ .

$\frac{y^{1} y^{2} - 1}{y^{2}}$

Step 4.1.1.2

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

$\frac{y^{1 + 2} - 1}{y^{2}}$

Step 4.1.2

Add $1$ and $2$ .

$\frac{y^{3} - 1}{y^{2}}$

Step 4.2

Rewrite $1$ as $1^{3}$ .

$\frac{y^{3} - 1^{3}}{y^{2}}$

Step 4.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = 1$ .

$\frac{(y - 1) (y^{2} + y \cdot 1 + 1^{2})}{y^{2}}$

Step 4.4

Simplify.

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Step 4.4.1

Multiply $y$ by $1$ .

$\frac{(y - 1) (y^{2} + y + 1^{2})}{y^{2}}$

Step 4.4.2

One to any power is one.

$\frac{(y - 1) (y^{2} + y + 1)}{y^{2}}$