Finite Math Examples

$\sqrt{\frac{75 x}{8}}$ , $\sqrt{\frac{3 x}{16}}$

Step 1

Add the expressions.

$\sqrt{\frac{75 x}{8}} + \sqrt{\frac{3 x}{16}}$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Rewrite $\frac{75 x}{8}$ as ${(\frac{5}{2})}^{2} \frac{3 x}{2}$ .

Tap for more steps...

Step 2.1.1

Factor the perfect power $5^{2}$ out of $75 x$ .

$\sqrt{\frac{5^{2} \cdot (3 x)}{8}} + \sqrt{\frac{3 x}{16}}$

Step 2.1.2

Factor the perfect power $2^{2}$ out of $8$ .

$\sqrt{\frac{5^{2} \cdot (3 x)}{2^{2} \cdot 2}} + \sqrt{\frac{3 x}{16}}$

Step 2.1.3

Rearrange the fraction $\frac{5^{2} \cdot (3 x)}{2^{2} \cdot 2}$ .

$\sqrt{{(\frac{5}{2})}^{2} \frac{3 x}{2}} + \sqrt{\frac{3 x}{16}}$

Step 2.2

Pull terms out from under the radical.

$\frac{5}{2} \sqrt{\frac{3 x}{2}} + \sqrt{\frac{3 x}{16}}$

Step 2.3

Rewrite $\sqrt{\frac{3 x}{2}}$ as $\frac{\sqrt{3 x}}{\sqrt{2}}$ .

$\frac{5}{2} \cdot \frac{\sqrt{3 x}}{\sqrt{2}} + \sqrt{\frac{3 x}{16}}$

Step 2.4

Multiply $\frac{\sqrt{3 x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{5}{2} (\frac{\sqrt{3 x}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}) + \sqrt{\frac{3 x}{16}}$

Step 2.5

Combine and simplify the denominator.

Tap for more steps...

Step 2.5.1

Multiply $\frac{\sqrt{3 x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{\sqrt{2} \sqrt{2}} + \sqrt{\frac{3 x}{16}}$

Step 2.5.2

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

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} + \sqrt{\frac{3 x}{16}}$

Step 2.5.3

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

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} + \sqrt{\frac{3 x}{16}}$

Step 2.5.4

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

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{1 + 1}} + \sqrt{\frac{3 x}{16}}$

Step 2.5.5

Add $1$ and $1$ .

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{{\sqrt{2}}^{2}} + \sqrt{\frac{3 x}{16}}$

Step 2.5.6

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

Tap for more steps...

Step 2.5.6.1

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

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} + \sqrt{\frac{3 x}{16}}$

Step 2.5.6.2

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

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} + \sqrt{\frac{3 x}{16}}$

Step 2.5.6.3

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

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2^{\frac{2}{2}}} + \sqrt{\frac{3 x}{16}}$

Step 2.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.5.6.4.1

Cancel the common factor.

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2^{\frac{2}{2}}} + \sqrt{\frac{3 x}{16}}$

Step 2.5.6.4.2

Rewrite the expression.

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2^{1}} + \sqrt{\frac{3 x}{16}}$

Step 2.5.6.5

Evaluate the exponent.

$\frac{5}{2} \cdot \frac{\sqrt{3 x} \sqrt{2}}{2} + \sqrt{\frac{3 x}{16}}$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Combine using the product rule for radicals.

$\frac{5}{2} \cdot \frac{\sqrt{3 x \cdot 2}}{2} + \sqrt{\frac{3 x}{16}}$

Step 2.6.2

Multiply $2$ by $3$ .

$\frac{5}{2} \cdot \frac{\sqrt{6 x}}{2} + \sqrt{\frac{3 x}{16}}$

Step 2.7

Multiply $\frac{5}{2} \cdot \frac{\sqrt{6 x}}{2}$ .

Tap for more steps...

Step 2.7.1

Multiply $\frac{5}{2}$ by $\frac{\sqrt{6 x}}{2}$ .

$\frac{5 \sqrt{6 x}}{2 \cdot 2} + \sqrt{\frac{3 x}{16}}$

Step 2.7.2

Multiply $2$ by $2$ .

$\frac{5 \sqrt{6 x}}{4} + \sqrt{\frac{3 x}{16}}$

Step 2.8

Rewrite $\frac{3 x}{16}$ as ${(\frac{1}{4})}^{2} (3 x)$ .

Tap for more steps...

Step 2.8.1

Factor the perfect power $1^{2}$ out of $3 x$ .

$\frac{5 \sqrt{6 x}}{4} + \sqrt{\frac{1^{2} (3 x)}{16}}$

Step 2.8.2

Factor the perfect power $4^{2}$ out of $16$ .

$\frac{5 \sqrt{6 x}}{4} + \sqrt{\frac{1^{2} (3 x)}{4^{2} \cdot 1}}$

Step 2.8.3

Rearrange the fraction $\frac{1^{2} (3 x)}{4^{2} \cdot 1}$ .

$\frac{5 \sqrt{6 x}}{4} + \sqrt{{(\frac{1}{4})}^{2} (3 x)}$

Step 2.9

Pull terms out from under the radical.

$\frac{5 \sqrt{6 x}}{4} + \frac{1}{4} \sqrt{3 x}$

Step 2.10

Combine $\frac{1}{4}$ and $\sqrt{3 x}$ .

$\frac{5 \sqrt{6 x}}{4} + \frac{\sqrt{3 x}}{4}$