Simplify the Radical Expression (5-4 square root of 5)(-2+ square root of 5)

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

Step 1

Expand $(5 - 4 \sqrt{5}) (- 2 + \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$5 \cdot - 2 + 5 \sqrt{5} - 4 \sqrt{5} \cdot - 2 - 4 \sqrt{5} \sqrt{5}$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $- 2$ .

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

Multiply $- 2$ by $- 4$ .

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 \sqrt{5} \sqrt{5}$

Multiply $- 4 \sqrt{5} \sqrt{5}$ .

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

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 ({\sqrt{5}}^{1} \sqrt{5})$

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

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

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

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 {\sqrt{5}}^{1 + 1}$

Add $1$ and $1$ .

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 {\sqrt{5}}^{2}$

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

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

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 {(5^{\frac{1}{2}})}^{2}$

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

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 \cdot 5^{\frac{1}{2} \cdot 2}$

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

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 \cdot 5^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 \cdot 5^{\frac{2}{2}}$

Rewrite the expression.

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 \cdot 5^{1}$

Evaluate the exponent.

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 4 \cdot 5$

Multiply $- 4$ by $5$ .

$- 10 + 5 \sqrt{5} + 8 \sqrt{5} - 20$

Subtract $20$ from $- 10$ .

$- 30 + 5 \sqrt{5} + 8 \sqrt{5}$

Add $5 \sqrt{5}$ and $8 \sqrt{5}$ .

$- 30 + 13 \sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 30 + 13 \sqrt{5}$

Decimal Form:

$- 0.93111629 \dots$

Simplify the Radical Expression (5-4 square root of 5)(-2+ square root of 5)

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