Simplify (5 square root of 3+ square root of 15)*(3 square root of 5- square root of 15)

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

Step 1

Expand $(5 \sqrt{3} + \sqrt{15}) (3 \sqrt{5} - \sqrt{15})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify each term.

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

Multiply $3$ by $5$ .

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

Combine using the product rule for radicals.

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

Multiply $5$ by $3$ .

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

Multiply $5 \sqrt{3} (- \sqrt{15})$ .

Multiply $- 1$ by $5$ .

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

Combine using the product rule for radicals.

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

Multiply $15$ by $3$ .

$15 \sqrt{15} - 5 \sqrt{45} + \sqrt{15} (3 \sqrt{5}) + \sqrt{15} (- \sqrt{15})$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$15 \sqrt{15} - 5 \sqrt{9 (5)} + \sqrt{15} (3 \sqrt{5}) + \sqrt{15} (- \sqrt{15})$

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

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

Pull terms out from under the radical.

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

Multiply $3$ by $- 5$ .

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

Multiply $\sqrt{15} (3 \sqrt{5})$ .

Combine using the product rule for radicals.

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

Multiply $15$ by $5$ .

$15 \sqrt{15} - 15 \sqrt{5} + 3 \sqrt{75} + \sqrt{15} (- \sqrt{15})$

Rewrite $75$ as $5^{2} \cdot 3$ .

Factor $25$ out of $75$ .

$15 \sqrt{15} - 15 \sqrt{5} + 3 \sqrt{25 (3)} + \sqrt{15} (- \sqrt{15})$

Rewrite $25$ as $5^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $5$ by $3$ .

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

Multiply $\sqrt{15} (- \sqrt{15})$ .

Raise $\sqrt{15}$ to the power of $1$ .

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

Raise $\sqrt{15}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$15 \sqrt{15} - 15 \sqrt{5} + 15 \sqrt{3} - 1 \cdot 15$

Multiply $- 1$ by $15$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$35.53449264 \dots$

Simplify (5 square root of 3+ square root of 15)*(3 square root of 5- square root of 15)

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