Evaluate square root of 5( square root of 45+ square root of 15)

$\sqrt{5} (\sqrt{45} + \sqrt{15})$

Step 1

Simplify each term.

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$\sqrt{5} (\sqrt{9 (5)} + \sqrt{15})$

Rewrite $9$ as $3^{2}$ .

$\sqrt{5} (\sqrt{3^{2} \cdot 5} + \sqrt{15})$

Pull terms out from under the radical.

$\sqrt{5} (3 \sqrt{5} + \sqrt{15})$

Step 2

Apply the distributive property.

$\sqrt{5} (3 \sqrt{5}) + \sqrt{5} \sqrt{15}$

Step 3

Multiply $\sqrt{5} (3 \sqrt{5})$ .

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

$3 ({\sqrt{5}}^{1} \sqrt{5}) + \sqrt{5} \sqrt{15}$

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

$3 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + \sqrt{5} \sqrt{15}$

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

$3 {\sqrt{5}}^{1 + 1} + \sqrt{5} \sqrt{15}$

Add $1$ and $1$ .

$3 {\sqrt{5}}^{2} + \sqrt{5} \sqrt{15}$

Step 4

Combine using the product rule for radicals.

$3 {\sqrt{5}}^{2} + \sqrt{5 \cdot 15}$

Step 5

Simplify each term.

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

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

$3 {(5^{\frac{1}{2}})}^{2} + \sqrt{5 \cdot 15}$

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

$3 \cdot 5^{\frac{1}{2} \cdot 2} + \sqrt{5 \cdot 15}$

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

$3 \cdot 5^{\frac{2}{2}} + \sqrt{5 \cdot 15}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 \cdot 5^{\frac{2}{2}} + \sqrt{5 \cdot 15}$

Rewrite the expression.

$3 \cdot 5^{1} + \sqrt{5 \cdot 15}$

Evaluate the exponent.

$3 \cdot 5 + \sqrt{5 \cdot 15}$

Multiply $3$ by $5$ .

$15 + \sqrt{5 \cdot 15}$

Multiply $5$ by $15$ .

$15 + \sqrt{75}$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$15 + \sqrt{25 (3)}$

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

$15 + \sqrt{5^{2} \cdot 3}$

Pull terms out from under the radical.

$15 + 5 \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$15 + 5 \sqrt{3}$

Decimal Form:

$23.66025403 \dots$

Evaluate square root of 5( square root of 45+ square root of 15)

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