Linear Algebra Examples

$\frac{20}{5} - \frac{1}{\sqrt{5}}$

Step 1

Simplify each term.

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Step 1.1

Divide $20$ by $5$ .

$4 - \frac{1}{\sqrt{5}}$

Step 1.2

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

$4 - (\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

$4 - \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 1.3.2

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

$4 - \frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 1.3.3

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

$4 - \frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 1.3.4

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

$4 - \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 1.3.5

Add $1$ and $1$ .

$4 - \frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 1.3.6

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

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Step 1.3.6.1

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

$4 - \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 1.3.6.2

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

$4 - \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 1.3.6.3

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

$4 - \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$4 - \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.3.6.4.2

Rewrite the expression.

$4 - \frac{\sqrt{5}}{5^{1}}$

Step 1.3.6.5

Evaluate the exponent.

$4 - \frac{\sqrt{5}}{5}$

Step 2

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$4 \cdot \frac{5}{5} - \frac{\sqrt{5}}{5}$

Step 3

Combine $4$ and $\frac{5}{5}$ .

$\frac{4 \cdot 5}{5} - \frac{\sqrt{5}}{5}$

Step 4

Simplify the expression.

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Step 4.1

Combine the numerators over the common denominator.

$\frac{4 \cdot 5 - \sqrt{5}}{5}$

Step 4.2

Multiply $4$ by $5$ .

$\frac{20 - \sqrt{5}}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{20 - \sqrt{5}}{5}$

Decimal Form:

$3.55278640 \dots$