Simplify ( square root of 3+ square root of 5x)( square root of 3-5 square root of 5x)

$(\sqrt{3} + \sqrt{5 x}) (\sqrt{3} - 5 \sqrt{5 x})$

Step 1

Expand $(\sqrt{3} + \sqrt{5 x}) (\sqrt{3} - 5 \sqrt{5 x})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{3} (\sqrt{3} - 5 \sqrt{5 x}) + \sqrt{5 x} (\sqrt{3} - 5 \sqrt{5 x})$

Apply the distributive property.

$\sqrt{3} \sqrt{3} + \sqrt{3} (- 5 \sqrt{5 x}) + \sqrt{5 x} (\sqrt{3} - 5 \sqrt{5 x})$

Apply the distributive property.

$\sqrt{3} \sqrt{3} + \sqrt{3} (- 5 \sqrt{5 x}) + \sqrt{5 x} \sqrt{3} + \sqrt{5 x} (- 5 \sqrt{5 x})$

Step 2

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\sqrt{3 \cdot 3} + \sqrt{3} (- 5 \sqrt{5 x}) + \sqrt{5 x} \sqrt{3} + \sqrt{5 x} (- 5 \sqrt{5 x})$

Multiply $3$ by $3$ .

$\sqrt{9} + \sqrt{3} (- 5 \sqrt{5 x}) + \sqrt{5 x} \sqrt{3} + \sqrt{5 x} (- 5 \sqrt{5 x})$

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

$3 + \sqrt{3} (- 5 \sqrt{5 x}) + \sqrt{5 x} \sqrt{3} + \sqrt{5 x} (- 5 \sqrt{5 x})$

Multiply $\sqrt{3} (- 5 \sqrt{5 x})$ .

Combine using the product rule for radicals.

$3 - 5 \sqrt{3 (5 x)} + \sqrt{5 x} \sqrt{3} + \sqrt{5 x} (- 5 \sqrt{5 x})$

Multiply $3$ by $5$ .

$3 - 5 \sqrt{15 x} + \sqrt{5 x} \sqrt{3} + \sqrt{5 x} (- 5 \sqrt{5 x})$

Multiply $\sqrt{5 x} \sqrt{3}$ .

Combine using the product rule for radicals.

$3 - 5 \sqrt{15 x} + \sqrt{5 x \cdot 3} + \sqrt{5 x} (- 5 \sqrt{5 x})$

Multiply $3$ by $5$ .

$3 - 5 \sqrt{15 x} + \sqrt{15 x} + \sqrt{5 x} (- 5 \sqrt{5 x})$

Rewrite using the commutative property of multiplication.

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 \sqrt{5 x} \sqrt{5 x}$

Multiply $- 5 \sqrt{5 x} \sqrt{5 x}$ .

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

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 ({\sqrt{5 x}}^{1} \sqrt{5 x})$

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

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 ({\sqrt{5 x}}^{1} {\sqrt{5 x}}^{1})$

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

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 {\sqrt{5 x}}^{1 + 1}$

Add $1$ and $1$ .

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 {\sqrt{5 x}}^{2}$

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

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

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 {({(5 x)}^{\frac{1}{2}})}^{2}$

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

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 {(5 x)}^{\frac{1}{2} \cdot 2}$

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

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 {(5 x)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 {(5 x)}^{\frac{2}{2}}$

Rewrite the expression.

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 {(5 x)}^{1}$

Simplify.

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 5 (5 x)$

Multiply $5$ by $- 5$ .

$3 - 5 \sqrt{15 x} + \sqrt{15 x} - 25 x$

Add $- 5 \sqrt{15 x}$ and $\sqrt{15 x}$ .

$3 - 4 \sqrt{15 x} - 25 x$

Step 3

Move $- 4 \sqrt{15 x}$ .

$3 - 25 x - 4 \sqrt{15 x}$

Step 4

Reorder $3$ and $- 25 x$ .

$- 25 x + 3 - 4 \sqrt{15 x}$

Simplify ( square root of 3+ square root of 5x)( square root of 3-5 square root of 5x)

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