Simplify the Radical Expression (5 square root of 5-2)( square root of 5-1)

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $5$ by $5$ .

$25 + 5 \sqrt{5} \cdot - 1 - 2 \sqrt{5} - 2 \cdot - 1$

Multiply $- 1$ by $5$ .

$25 - 5 \sqrt{5} - 2 \sqrt{5} - 2 \cdot - 1$

Multiply $- 2$ by $- 1$ .

$25 - 5 \sqrt{5} - 2 \sqrt{5} + 2$

Add $25$ and $2$ .

$27 - 5 \sqrt{5} - 2 \sqrt{5}$

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

$27 - 7 \sqrt{5}$

The result can be shown in multiple forms.

Exact Form:

$27 - 7 \sqrt{5}$

Decimal Form:

$11.34752415 \dots$

Simplify the Radical Expression (5 square root of 5-2)( square root of 5-1)

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