Simplify 8 square root of 5*( square root of 10+ square root of 5)

$8 \sqrt{5} \cdot (\sqrt{10} + \sqrt{5})$

Step 1

Apply the distributive property.

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

Step 2

Multiply $8 \sqrt{5} \sqrt{10}$ .

Combine using the product rule for radicals.

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

Multiply $10$ by $5$ .

$8 \sqrt{50} + 8 \sqrt{5} \sqrt{5}$

Step 3

Multiply $8 \sqrt{5} \sqrt{5}$ .

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

$8 \sqrt{50} + 8 ({\sqrt{5}}^{1} \sqrt{5})$

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

$8 \sqrt{50} + 8 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

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

$8 \sqrt{50} + 8 {\sqrt{5}}^{1 + 1}$

Add $1$ and $1$ .

$8 \sqrt{50} + 8 {\sqrt{5}}^{2}$

Step 4

Simplify each term.

Rewrite $50$ as $5^{2} \cdot 2$ .

Factor $25$ out of $50$ .

$8 \sqrt{25 (2)} + 8 {\sqrt{5}}^{2}$

Rewrite $25$ as $5^{2}$ .

$8 \sqrt{5^{2} \cdot 2} + 8 {\sqrt{5}}^{2}$

Pull terms out from under the radical.

$8 (5 \sqrt{2}) + 8 {\sqrt{5}}^{2}$

Multiply $5$ by $8$ .

$40 \sqrt{2} + 8 {\sqrt{5}}^{2}$

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

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

$40 \sqrt{2} + 8 {(5^{\frac{1}{2}})}^{2}$

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

$40 \sqrt{2} + 8 \cdot 5^{\frac{1}{2} \cdot 2}$

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

$40 \sqrt{2} + 8 \cdot 5^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$40 \sqrt{2} + 8 \cdot 5^{\frac{2}{2}}$

Rewrite the expression.

$40 \sqrt{2} + 8 \cdot 5^{1}$

Evaluate the exponent.

$40 \sqrt{2} + 8 \cdot 5$

Multiply $8$ by $5$ .

$40 \sqrt{2} + 40$

Step 5

The result can be shown in multiple forms.

Exact Form:

$40 \sqrt{2} + 40$

Decimal Form:

$96.56854249 \dots$

Simplify 8 square root of 5*( square root of 10+ square root of 5)

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