Pre-Algebra Examples

$\frac{2 \sqrt{5} - 3}{\sqrt{5} - 1}$

Step 1

Multiply $\frac{2 \sqrt{5} - 3}{\sqrt{5} - 1}$ by $\frac{\sqrt{5} + 1}{\sqrt{5} + 1}$ .

$\frac{2 \sqrt{5} - 3}{\sqrt{5} - 1} \cdot \frac{\sqrt{5} + 1}{\sqrt{5} + 1}$

Step 2

Multiply $\frac{2 \sqrt{5} - 3}{\sqrt{5} - 1}$ by $\frac{\sqrt{5} + 1}{\sqrt{5} + 1}$ .

$\frac{(2 \sqrt{5} - 3) (\sqrt{5} + 1)}{(\sqrt{5} - 1) (\sqrt{5} + 1)}$

Step 3

Expand the denominator using the FOIL method.

$\frac{(2 \sqrt{5} - 3) (\sqrt{5} + 1)}{{\sqrt{5}}^{2} + \sqrt{5} - \sqrt{5} - 1}$

Step 4

Simplify.

$\frac{(2 \sqrt{5} - 3) (\sqrt{5} + 1)}{4}$

Step 5

Expand $(2 \sqrt{5} - 3) (\sqrt{5} + 1)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\frac{2 \sqrt{5} (\sqrt{5} + 1) - 3 (\sqrt{5} + 1)}{4}$

Step 5.2

Apply the distributive property.

$\frac{2 \sqrt{5} \sqrt{5} + 2 \sqrt{5} \cdot 1 - 3 (\sqrt{5} + 1)}{4}$

Step 5.3

Apply the distributive property.

$\frac{2 \sqrt{5} \sqrt{5} + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $2 \sqrt{5} \sqrt{5}$ .

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Step 6.1.1.1

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

$\frac{2 ({\sqrt{5}}^{1} \sqrt{5}) + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.1.2

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

$\frac{2 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.1.3

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

$\frac{2 {\sqrt{5}}^{1 + 1} + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.1.4

Add $1$ and $1$ .

$\frac{2 {\sqrt{5}}^{2} + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.2

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

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Step 6.1.2.1

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

$\frac{2 {(5^{\frac{1}{2}})}^{2} + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.2.2

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

$\frac{2 \cdot 5^{\frac{1}{2} \cdot 2} + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.2.3

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

$\frac{2 \cdot 5^{\frac{2}{2}} + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.2.4

Cancel the common factor of $2$ .

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Step 6.1.2.4.1

Cancel the common factor.

$\frac{2 \cdot 5^{\frac{2}{2}} + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.2.4.2

Rewrite the expression.

$\frac{2 \cdot 5^{1} + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.2.5

Evaluate the exponent.

$\frac{2 \cdot 5 + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.3

Multiply $2$ by $5$ .

$\frac{10 + 2 \sqrt{5} \cdot 1 - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.4

Multiply $2$ by $1$ .

$\frac{10 + 2 \sqrt{5} - 3 \sqrt{5} - 3 \cdot 1}{4}$

Step 6.1.5

Multiply $- 3$ by $1$ .

$\frac{10 + 2 \sqrt{5} - 3 \sqrt{5} - 3}{4}$

Step 6.2

Subtract $3$ from $10$ .

$\frac{7 + 2 \sqrt{5} - 3 \sqrt{5}}{4}$

Step 6.3

Subtract $3 \sqrt{5}$ from $2 \sqrt{5}$ .

$\frac{7 - \sqrt{5}}{4}$

Step 7

Split the fraction $\frac{7 - \sqrt{5}}{4}$ into two fractions.

$\frac{7}{4} + \frac{- \sqrt{5}}{4}$

Step 8

Move the negative in front of the fraction.

$\frac{7}{4} - \frac{\sqrt{5}}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{4} - \frac{\sqrt{5}}{4}$

Decimal Form:

$1.19098300 \dots$