Simplify ( square root of 75a+ square root of 12a- square root of 27a)/( square root of 3a)

$\frac{\sqrt{75 a} + \sqrt{12 a} - \sqrt{27 a}}{\sqrt{3 a}}$

Step 1

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{75 a}$ as ${(75 a)}^{\frac{1}{2}}$ .

$\frac{{(75 a)}^{\frac{1}{2}} + \sqrt{12 a} - \sqrt{27 a}}{\sqrt{3 a}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12 a}$ as ${(12 a)}^{\frac{1}{2}}$ .

$\frac{{(75 a)}^{\frac{1}{2}} + {(12 a)}^{\frac{1}{2}} - \sqrt{27 a}}{\sqrt{3 a}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{27 a}$ as ${(27 a)}^{\frac{1}{2}}$ .

$\frac{{(75 a)}^{\frac{1}{2}} + {(12 a)}^{\frac{1}{2}} - {(27 a)}^{\frac{1}{2}}}{\sqrt{3 a}}$

Apply the product rule to $75 a$ .

$\frac{75^{\frac{1}{2}} a^{\frac{1}{2}} + {(12 a)}^{\frac{1}{2}} - {(27 a)}^{\frac{1}{2}}}{\sqrt{3 a}}$

Apply the product rule to $12 a$ .

$\frac{75^{\frac{1}{2}} a^{\frac{1}{2}} + 12^{\frac{1}{2}} a^{\frac{1}{2}} - {(27 a)}^{\frac{1}{2}}}{\sqrt{3 a}}$

Apply the product rule to $27 a$ .

$\frac{75^{\frac{1}{2}} a^{\frac{1}{2}} + 12^{\frac{1}{2}} a^{\frac{1}{2}} - (27^{\frac{1}{2}} a^{\frac{1}{2}})}{\sqrt{3 a}}$

Factor $a^{\frac{1}{2}}$ out of $75^{\frac{1}{2}} a^{\frac{1}{2}} + 12^{\frac{1}{2}} a^{\frac{1}{2}} - 27^{\frac{1}{2}} a^{\frac{1}{2}}$ .

Factor $a^{\frac{1}{2}}$ out of $75^{\frac{1}{2}} a^{\frac{1}{2}}$ .

$\frac{a^{\frac{1}{2}} \cdot 75^{\frac{1}{2}} + 12^{\frac{1}{2}} a^{\frac{1}{2}} - 27^{\frac{1}{2}} a^{\frac{1}{2}}}{\sqrt{3 a}}$

Factor $a^{\frac{1}{2}}$ out of $12^{\frac{1}{2}} a^{\frac{1}{2}}$ .

$\frac{a^{\frac{1}{2}} \cdot 75^{\frac{1}{2}} + a^{\frac{1}{2}} \cdot 12^{\frac{1}{2}} - 27^{\frac{1}{2}} a^{\frac{1}{2}}}{\sqrt{3 a}}$

Factor $a^{\frac{1}{2}}$ out of $- 27^{\frac{1}{2}} a^{\frac{1}{2}}$ .

$\frac{a^{\frac{1}{2}} \cdot 75^{\frac{1}{2}} + a^{\frac{1}{2}} \cdot 12^{\frac{1}{2}} + a^{\frac{1}{2}} (- 27^{\frac{1}{2}})}{\sqrt{3 a}}$

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} \cdot 75^{\frac{1}{2}} + a^{\frac{1}{2}} \cdot 12^{\frac{1}{2}}$ .

$\frac{a^{\frac{1}{2}} \cdot (75^{\frac{1}{2}} + 12^{\frac{1}{2}}) + a^{\frac{1}{2}} (- 27^{\frac{1}{2}})}{\sqrt{3 a}}$

Factor $a^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} \cdot (75^{\frac{1}{2}} + 12^{\frac{1}{2}}) + a^{\frac{1}{2}} (- 27^{\frac{1}{2}})$ .

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}})}{\sqrt{3 a}}$

Step 2

Multiply $\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}})}{\sqrt{3 a}}$ by $\frac{\sqrt{3 a}}{\sqrt{3 a}}$ .

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}})}{\sqrt{3 a}} \cdot \frac{\sqrt{3 a}}{\sqrt{3 a}}$

Step 3

Combine and simplify the denominator.

Multiply $\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}})}{\sqrt{3 a}}$ by $\frac{\sqrt{3 a}}{\sqrt{3 a}}$ .

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{\sqrt{3 a} \sqrt{3 a}}$

Raise $\sqrt{3 a}$ to the power of $1$ .

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{{\sqrt{3 a}}^{1} \sqrt{3 a}}$

Raise $\sqrt{3 a}$ to the power of $1$ .

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{{\sqrt{3 a}}^{1} {\sqrt{3 a}}^{1}}$

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

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{{\sqrt{3 a}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{{\sqrt{3 a}}^{2}}$

Rewrite ${\sqrt{3 a}}^{2}$ as $3 a$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 a}$ as ${(3 a)}^{\frac{1}{2}}$ .

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{{({(3 a)}^{\frac{1}{2}})}^{2}}$

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

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{{(3 a)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{{(3 a)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{{(3 a)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{{(3 a)}^{1}}$

Simplify.

$\frac{a^{\frac{1}{2}} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{3 a}$

Step 4

Move $a^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{(75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{3 a \cdot a^{- \frac{1}{2}}}$

Step 5

Multiply $a$ by $a^{- \frac{1}{2}}$ by adding the exponents.

Move $a^{- \frac{1}{2}}$ .

$\frac{(75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{3 (a^{- \frac{1}{2}} a)}$

Multiply $a^{- \frac{1}{2}}$ by $a$ .

Raise $a$ to the power of $1$ .

$\frac{(75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{3 (a^{- \frac{1}{2}} a^{1})}$

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

$\frac{(75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{3 a^{- \frac{1}{2} + 1}}$

Write $1$ as a fraction with a common denominator.

$\frac{(75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{3 a^{- \frac{1}{2} + \frac{2}{2}}}$

Combine the numerators over the common denominator.

$\frac{(75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{3 a^{\frac{- 1 + 2}{2}}}$

Add $- 1$ and $2$ .

$\frac{(75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{3 a^{\frac{1}{2}}}$

Step 6

Reorder factors in $\frac{(75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}}) \sqrt{3 a}}{3 a^{\frac{1}{2}}}$ .

$\frac{\sqrt{3 a} (75^{\frac{1}{2}} + 12^{\frac{1}{2}} - 27^{\frac{1}{2}})}{3 a^{\frac{1}{2}}}$

Simplify ( square root of 75a+ square root of 12a- square root of 27a)/( square root of 3a)

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