Simplify the Radical Expression (4 square root of 5-1)(3+2 square root of 5)

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

Step 1

Expand $(4 \sqrt{5} - 1) (3 + 2 \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $4$ .

$12 \sqrt{5} + 4 \sqrt{5} (2 \sqrt{5}) - 1 \cdot 3 - 1 (2 \sqrt{5})$

Multiply $4 \sqrt{5} (2 \sqrt{5})$ .

Multiply $2$ by $4$ .

$12 \sqrt{5} + 8 \sqrt{5} \sqrt{5} - 1 \cdot 3 - 1 (2 \sqrt{5})$

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

$12 \sqrt{5} + 8 ({\sqrt{5}}^{1} \sqrt{5}) - 1 \cdot 3 - 1 (2 \sqrt{5})$

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

$12 \sqrt{5} + 8 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) - 1 \cdot 3 - 1 (2 \sqrt{5})$

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

$12 \sqrt{5} + 8 {\sqrt{5}}^{1 + 1} - 1 \cdot 3 - 1 (2 \sqrt{5})$

Add $1$ and $1$ .

$12 \sqrt{5} + 8 {\sqrt{5}}^{2} - 1 \cdot 3 - 1 (2 \sqrt{5})$

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

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

$12 \sqrt{5} + 8 {(5^{\frac{1}{2}})}^{2} - 1 \cdot 3 - 1 (2 \sqrt{5})$

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

$12 \sqrt{5} + 8 \cdot 5^{\frac{1}{2} \cdot 2} - 1 \cdot 3 - 1 (2 \sqrt{5})$

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

$12 \sqrt{5} + 8 \cdot 5^{\frac{2}{2}} - 1 \cdot 3 - 1 (2 \sqrt{5})$

Cancel the common factor of $2$ .

Cancel the common factor.

$12 \sqrt{5} + 8 \cdot 5^{\frac{2}{2}} - 1 \cdot 3 - 1 (2 \sqrt{5})$

Rewrite the expression.

$12 \sqrt{5} + 8 \cdot 5^{1} - 1 \cdot 3 - 1 (2 \sqrt{5})$

Evaluate the exponent.

$12 \sqrt{5} + 8 \cdot 5 - 1 \cdot 3 - 1 (2 \sqrt{5})$

Multiply $8$ by $5$ .

$12 \sqrt{5} + 40 - 1 \cdot 3 - 1 (2 \sqrt{5})$

Multiply $- 1$ by $3$ .

$12 \sqrt{5} + 40 - 3 - 1 (2 \sqrt{5})$

Multiply $2$ by $- 1$ .

$12 \sqrt{5} + 40 - 3 - 2 \sqrt{5}$

Subtract $2 \sqrt{5}$ from $12 \sqrt{5}$ .

$40 - 3 + 10 \sqrt{5}$

Subtract $3$ from $40$ .

$37 + 10 \sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$37 + 10 \sqrt{5}$

Decimal Form:

$59.36067977 \dots$

Simplify the Radical Expression (4 square root of 5-1)(3+2 square root of 5)

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