Basic Math Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Combine $\sqrt{2 n^{2}}$ and $\sqrt{10 n}$ into a single radical.

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

Step 1.2

Reduce the expression $\frac{2 n^{2}}{10 n}$ by cancelling the common factors.

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

Factor $2$ out of $10 n$ .

$3 n^{2} + \sqrt{\frac{2 (n^{2})}{2 (5 n)}}$

Step 1.2.3

Cancel the common factor.

$3 n^{2} + \sqrt{\frac{2 n^{2}}{2 (5 n)}}$

Step 1.2.4

Rewrite the expression.

$3 n^{2} + \sqrt{\frac{n^{2}}{5 n}}$

Step 1.3

Cancel the common factor of $n^{2}$ and $n$ .

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

Cancel the common factors.

Tap for more steps...

Step 1.3.2.1

Factor $n$ out of $5 n$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

$3 n^{2} + \sqrt{\frac{n}{5}}$

Step 1.4

Rewrite $\sqrt{\frac{n}{5}}$ as $\frac{\sqrt{n}}{\sqrt{5}}$ .

$3 n^{2} + \frac{\sqrt{n}}{\sqrt{5}}$

Step 1.5

Multiply $\frac{\sqrt{n}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

Step 1.6

Combine and simplify the denominator.

Tap for more steps...

Step 1.6.1

Multiply $\frac{\sqrt{n}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$3 n^{2} + \frac{\sqrt{n} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 1.6.2

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

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

Step 1.6.3

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

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

Step 1.6.4

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

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

Step 1.6.5

Add $1$ and $1$ .

$3 n^{2} + \frac{\sqrt{n} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 1.6.6

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

Tap for more steps...

Step 1.6.6.1

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

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

Step 1.6.6.2

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

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

Step 1.6.6.3

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

$3 n^{2} + \frac{\sqrt{n} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.6.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.6.6.4.1

Cancel the common factor.

$3 n^{2} + \frac{\sqrt{n} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.6.6.4.2

Rewrite the expression.

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

Step 1.6.6.5

Evaluate the exponent.

$3 n^{2} + \frac{\sqrt{n} \sqrt{5}}{5}$

Step 1.7

Combine using the product rule for radicals.

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

Step 2

To write $3 n^{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Combine $3 n^{2}$ and $\frac{5}{5}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Multiply $5$ by $3$ .

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

Step 4.3

Move $5$ to the left of $n$ .

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

Step 4.4

Apply the product rule to $5 n$ .

$\frac{15 n^{2} + 5^{\frac{1}{2}} n^{\frac{1}{2}}}{5}$

Step 4.5

Factor $n^{\frac{1}{2}}$ out of $15 n^{2} + 5^{\frac{1}{2}} n^{\frac{1}{2}}$ .

Tap for more steps...

Step 4.5.1

Factor $n^{\frac{1}{2}}$ out of $15 n^{2}$ .

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

Step 4.5.2

Factor $n^{\frac{1}{2}}$ out of $5^{\frac{1}{2}} n^{\frac{1}{2}}$ .

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

Step 4.5.3

Factor $n^{\frac{1}{2}}$ out of $n^{\frac{1}{2}} (15 n^{\frac{3}{2}}) + n^{\frac{1}{2}} \cdot 5^{\frac{1}{2}}$ .

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