Simplify ( square root of 5- square root of 8)(4 square root of 5+9 square root of 8)

$(\sqrt{5} - \sqrt{8}) (4 \sqrt{5} + 9 \sqrt{8})$

Step 1

Simplify terms.

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$(\sqrt{5} - \sqrt{4 (2)}) (4 \sqrt{5} + 9 \sqrt{8})$

Rewrite $4$ as $2^{2}$ .

$(\sqrt{5} - \sqrt{2^{2} \cdot 2}) (4 \sqrt{5} + 9 \sqrt{8})$

Pull terms out from under the radical.

$(\sqrt{5} - (2 \sqrt{2})) (4 \sqrt{5} + 9 \sqrt{8})$

Multiply $2$ by $- 1$ .

$(\sqrt{5} - 2 \sqrt{2}) (4 \sqrt{5} + 9 \sqrt{8})$

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$(\sqrt{5} - 2 \sqrt{2}) (4 \sqrt{5} + 9 \sqrt{4 (2)})$

Rewrite $4$ as $2^{2}$ .

$(\sqrt{5} - 2 \sqrt{2}) (4 \sqrt{5} + 9 \sqrt{2^{2} \cdot 2})$

Pull terms out from under the radical.

$(\sqrt{5} - 2 \sqrt{2}) (4 \sqrt{5} + 9 (2 \sqrt{2}))$

Multiply $2$ by $9$ .

$(\sqrt{5} - 2 \sqrt{2}) (4 \sqrt{5} + 18 \sqrt{2})$

Step 2

Expand $(\sqrt{5} - 2 \sqrt{2}) (4 \sqrt{5} + 18 \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{5} (4 \sqrt{5} + 18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5} + 18 \sqrt{2})$

Apply the distributive property.

$\sqrt{5} (4 \sqrt{5}) + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5} + 18 \sqrt{2})$

Apply the distributive property.

$\sqrt{5} (4 \sqrt{5}) + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

Step 3

Simplify and combine like terms.

Simplify each term.

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

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

$4 ({\sqrt{5}}^{1} \sqrt{5}) + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

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

$4 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

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

$4 {\sqrt{5}}^{1 + 1} + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

Add $1$ and $1$ .

$4 {\sqrt{5}}^{2} + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

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

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

$4 {(5^{\frac{1}{2}})}^{2} + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

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

$4 \cdot 5^{\frac{1}{2} \cdot 2} + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

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

$4 \cdot 5^{\frac{2}{2}} + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 \cdot 5^{\frac{2}{2}} + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

Rewrite the expression.

$4 \cdot 5^{1} + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

Evaluate the exponent.

$4 \cdot 5 + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

Multiply $4$ by $5$ .

$20 + \sqrt{5} (18 \sqrt{2}) - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

Multiply $\sqrt{5} (18 \sqrt{2})$ .

Combine using the product rule for radicals.

$20 + 18 \sqrt{5 \cdot 2} - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

Multiply $5$ by $2$ .

$20 + 18 \sqrt{10} - 2 \sqrt{2} (4 \sqrt{5}) - 2 \sqrt{2} (18 \sqrt{2})$

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

Multiply $4$ by $- 2$ .

$20 + 18 \sqrt{10} - 8 \sqrt{2} \sqrt{5} - 2 \sqrt{2} (18 \sqrt{2})$

Combine using the product rule for radicals.

$20 + 18 \sqrt{10} - 8 \sqrt{5 \cdot 2} - 2 \sqrt{2} (18 \sqrt{2})$

Multiply $5$ by $2$ .

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 2 \sqrt{2} (18 \sqrt{2})$

Multiply $- 2 \sqrt{2} (18 \sqrt{2})$ .

Multiply $18$ by $- 2$ .

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 \sqrt{2} \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 ({\sqrt{2}}^{1} \sqrt{2})$

Raise $\sqrt{2}$ to the power of $1$ .

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

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

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 {\sqrt{2}}^{2}$

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

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

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 {(2^{\frac{1}{2}})}^{2}$

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

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 \cdot 2^{\frac{1}{2} \cdot 2}$

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

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 \cdot 2^{1}$

Evaluate the exponent.

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 36 \cdot 2$

Multiply $- 36$ by $2$ .

$20 + 18 \sqrt{10} - 8 \sqrt{10} - 72$

Subtract $72$ from $20$ .

$- 52 + 18 \sqrt{10} - 8 \sqrt{10}$

Subtract $8 \sqrt{10}$ from $18 \sqrt{10}$ .

$- 52 + 10 \sqrt{10}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$- 52 + 10 \sqrt{10}$

Decimal Form:

$- 20.37722339 \dots$

Simplify ( square root of 5- square root of 8)(4 square root of 5+9 square root of 8)

Related Questions

Support LCMGCF.com by disabling your adblocker.