Simplify (5(a+ square root of 2))/(2a- square root of 3)

$\frac{5 (a + \sqrt{2})}{2 a - \sqrt{3}}$

Step 1

Multiply $\frac{5 (a + \sqrt{2})}{2 a - \sqrt{3}}$ by $\frac{2 a + \sqrt{3}}{2 a + \sqrt{3}}$ .

$\frac{5 (a + \sqrt{2})}{2 a - \sqrt{3}} \cdot \frac{2 a + \sqrt{3}}{2 a + \sqrt{3}}$

Step 2

Combine fractions.

Multiply $\frac{5 (a + \sqrt{2})}{2 a - \sqrt{3}}$ by $\frac{2 a + \sqrt{3}}{2 a + \sqrt{3}}$ .

$\frac{5 (a + \sqrt{2}) (2 a + \sqrt{3})}{(2 a - \sqrt{3}) (2 a + \sqrt{3})}$

Expand the denominator using the FOIL method.

$\frac{5 (a + \sqrt{2}) (2 a + \sqrt{3})}{4 a^{2} + 2 a \sqrt{3} - 2 \sqrt{3} a - {\sqrt{3}}^{2}}$

Simplify.

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

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

$\frac{5 (a + \sqrt{2}) (2 a + \sqrt{3})}{4 a^{2} - {(3^{\frac{1}{2}})}^{2}}$

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

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

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

$\frac{5 (a + \sqrt{2}) (2 a + \sqrt{3})}{4 a^{2} - 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{5 (a + \sqrt{2}) (2 a + \sqrt{3})}{4 a^{2} - 3^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{5 (a + \sqrt{2}) (2 a + \sqrt{3})}{4 a^{2} - 3^{1}}$

Evaluate the exponent.

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

Multiply $- 1$ by $3$ .

$\frac{5 (a + \sqrt{2}) (2 a + \sqrt{3})}{4 a^{2} - 3}$

Group $2 a + \sqrt{3}$ and $a + \sqrt{2}$ together.

$\frac{5 ((2 a + \sqrt{3}) (a + \sqrt{2}))}{4 a^{2} - 3}$

Step 3

Expand $(2 a + \sqrt{3}) (a + \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\frac{5 (2 a (a + \sqrt{2}) + \sqrt{3} (a + \sqrt{2}))}{4 a^{2} - 3}$

Apply the distributive property.

$\frac{5 (2 a \cdot a + 2 a \sqrt{2} + \sqrt{3} (a + \sqrt{2}))}{4 a^{2} - 3}$

Apply the distributive property.

$\frac{5 (2 a \cdot a + 2 a \sqrt{2} + \sqrt{3} a + \sqrt{3} \sqrt{2})}{4 a^{2} - 3}$

Step 4

Simplify terms.

Simplify each term.

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$\frac{5 (2 (a \cdot a) + 2 a \sqrt{2} + \sqrt{3} a + \sqrt{3} \sqrt{2})}{4 a^{2} - 3}$

Multiply $a$ by $a$ .

$\frac{5 (2 a^{2} + 2 a \sqrt{2} + \sqrt{3} a + \sqrt{3} \sqrt{2})}{4 a^{2} - 3}$

Combine using the product rule for radicals.

$\frac{5 (2 a^{2} + 2 a \sqrt{2} + \sqrt{3} a + \sqrt{3 \cdot 2})}{4 a^{2} - 3}$

Multiply $3$ by $2$ .

$\frac{5 (2 a^{2} + 2 a \sqrt{2} + \sqrt{3} a + \sqrt{6})}{4 a^{2} - 3}$

Reorder terms.

$\frac{5 (2 a^{2} + 2 a \sqrt{2} + a \sqrt{3} + \sqrt{6})}{4 a^{2} - 3}$

Simplify (5(a+ square root of 2))/(2a- square root of 3)

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