Basic Math Examples

$\frac{1}{\sqrt{5}} - \frac{\sqrt{5}}{\sqrt{5}}$

Step 1

Combine the numerators over the common denominator.

$\frac{1 - \sqrt{5}}{\sqrt{5}}$

Step 2

Multiply $\frac{1 - \sqrt{5}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1 - \sqrt{5}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{1 - \sqrt{5}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{(1 - \sqrt{5}) \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{(1 - \sqrt{5}) \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 3.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{(1 - \sqrt{5}) \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 3.4

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

$\frac{(1 - \sqrt{5}) \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{(1 - \sqrt{5}) \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{(1 - \sqrt{5}) \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$\frac{(1 - \sqrt{5}) \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$\frac{(1 - \sqrt{5}) \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{(1 - \sqrt{5}) \sqrt{5}}{5^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{(1 - \sqrt{5}) \sqrt{5}}{5^{1}}$

Step 3.6.5

Evaluate the exponent.

$\frac{(1 - \sqrt{5}) \sqrt{5}}{5}$

Step 4

Apply the distributive property.

$\frac{1 \sqrt{5} - \sqrt{5} \sqrt{5}}{5}$

Step 5

Multiply $\sqrt{5}$ by $1$ .

$\frac{\sqrt{5} - \sqrt{5} \sqrt{5}}{5}$

Step 6

Multiply $- \sqrt{5} \sqrt{5}$ .

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Step 6.1

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{5} - ({\sqrt{5}}^{1} \sqrt{5})}{5}$

Step 6.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{5} - ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}{5}$

Step 6.3

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

$\frac{\sqrt{5} - {\sqrt{5}}^{1 + 1}}{5}$

Step 6.4

Add $1$ and $1$ .

$\frac{\sqrt{5} - {\sqrt{5}}^{2}}{5}$

Step 7

Simplify each term.

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Step 7.1

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{5} - {(5^{\frac{1}{2}})}^{2}}{5}$

Step 7.1.2

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

$\frac{\sqrt{5} - 5^{\frac{1}{2} \cdot 2}}{5}$

Step 7.1.3

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

$\frac{\sqrt{5} - 5^{\frac{2}{2}}}{5}$

Step 7.1.4

Cancel the common factor of $2$ .

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Step 7.1.4.1

Cancel the common factor.

$\frac{\sqrt{5} - 5^{\frac{2}{2}}}{5}$

Step 7.1.4.2

Rewrite the expression.

$\frac{\sqrt{5} - 5^{1}}{5}$

Step 7.1.5

Evaluate the exponent.

$\frac{\sqrt{5} - 1 \cdot 5}{5}$

Step 7.2

Multiply $- 1$ by $5$ .

$\frac{\sqrt{5} - 5}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5} - 5}{5}$

Decimal Form:

$- 0.55278640 \dots$