Divide (3n^2+ square root of 2n^2)/( square root of 10n)

$\frac{3 n^{2} + \sqrt{2 n^{2}}}{\sqrt{10 n}}$

Simplify the numerator.

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

$\frac{3 n^{2} + {(2 n^{2})}^{\frac{1}{2}}}{\sqrt{10 n}}$

Apply the product rule to $2 n^{2}$ .

$\frac{3 n^{2} + 2^{\frac{1}{2}} {(n^{2})}^{\frac{1}{2}}}{\sqrt{10 n}}$

Multiply the exponents in ${(n^{2})}^{\frac{1}{2}}$ .

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

$\frac{3 n^{2} + 2^{\frac{1}{2}} n^{2 (\frac{1}{2})}}{\sqrt{10 n}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 n^{2} + 2^{\frac{1}{2}} n^{2 (\frac{1}{2})}}{\sqrt{10 n}}$

Rewrite the expression.

$\frac{3 n^{2} + 2^{\frac{1}{2}} n^{1}}{\sqrt{10 n}}$

Simplify.

$\frac{3 n^{2} + 2^{\frac{1}{2}} n}{\sqrt{10 n}}$

Factor $n$ out of $3 n^{2} + 2^{\frac{1}{2}} n$ .

Factor $n$ out of $3 n^{2}$ .

$\frac{n (3 n) + 2^{\frac{1}{2}} n}{\sqrt{10 n}}$

Factor $n$ out of $2^{\frac{1}{2}} n$ .

$\frac{n (3 n) + n \cdot 2^{\frac{1}{2}}}{\sqrt{10 n}}$

Factor $n$ out of $n (3 n) + n \cdot 2^{\frac{1}{2}}$ .

$\frac{n (3 n + 2^{\frac{1}{2}})}{\sqrt{10 n}}$

Multiply $\frac{n (3 n + 2^{\frac{1}{2}})}{\sqrt{10 n}}$ by $\frac{\sqrt{10 n}}{\sqrt{10 n}}$ .

$\frac{n (3 n + 2^{\frac{1}{2}})}{\sqrt{10 n}} \cdot \frac{\sqrt{10 n}}{\sqrt{10 n}}$

Combine and simplify the denominator.

Multiply $\frac{n (3 n + 2^{\frac{1}{2}})}{\sqrt{10 n}}$ by $\frac{\sqrt{10 n}}{\sqrt{10 n}}$ .

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{\sqrt{10 n} \sqrt{10 n}}$

Raise $\sqrt{10 n}$ to the power of $1$ .

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{\sqrt{10 n}}^{1} \sqrt{10 n}}$

Raise $\sqrt{10 n}$ to the power of $1$ .

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{\sqrt{10 n}}^{1} {\sqrt{10 n}}^{1}}$

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

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{\sqrt{10 n}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{\sqrt{10 n}}^{2}}$

Rewrite ${\sqrt{10 n}}^{2}$ as $10 n$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 n}$ as ${(10 n)}^{\frac{1}{2}}$ .

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{({(10 n)}^{\frac{1}{2}})}^{2}}$

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

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{(10 n)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{(10 n)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{(10 n)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{{(10 n)}^{1}}$

Simplify.

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{10 n}$

Cancel the common factor.

$\frac{n (3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{10 n}$

Rewrite the expression.

$\frac{(3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{10}$

Reorder factors in $\frac{(3 n + 2^{\frac{1}{2}}) \sqrt{10 n}}{10}$ .

$\frac{\sqrt{10 n} (3 n + 2^{\frac{1}{2}})}{10}$

Apply the distributive property.

$\frac{\sqrt{10 n} (3 n) + \sqrt{10 n} \cdot 2^{\frac{1}{2}}}{10}$

Rewrite using the commutative property of multiplication.

$\frac{3 \sqrt{10 n} n + \sqrt{10 n} \cdot 2^{\frac{1}{2}}}{10}$

Split the fraction $\frac{3 \sqrt{10 n} n + \sqrt{10 n} \cdot 2^{\frac{1}{2}}}{10}$ into two fractions.

$\frac{3 \sqrt{10 n} n}{10} + \frac{\sqrt{10 n} \cdot 2^{\frac{1}{2}}}{10}$

Divide (3n^2+ square root of 2n^2)/( square root of 10n)

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