Simplify the Radical Expression ( square root of 5- square root of 3)^2

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

Step 1

Rewrite ${(\sqrt{5} - \sqrt{3})}^{2}$ as $(\sqrt{5} - \sqrt{3}) (\sqrt{5} - \sqrt{3})$ .

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $5$ by $5$ .

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

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

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

Pull terms out from under the radical.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

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

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

Combine using the product rule for radicals.

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

Multiply $5$ by $3$ .

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

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

Combine using the product rule for radicals.

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

Multiply $5$ by $3$ .

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $\sqrt{3}$ by $1$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Add $5$ and $3$ .

$8 - \sqrt{15} - \sqrt{15}$

Subtract $\sqrt{15}$ from $- \sqrt{15}$ .

$8 - 2 \sqrt{15}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$8 - 2 \sqrt{15}$

Decimal Form:

$0.25403330 \dots$

Simplify the Radical Expression ( square root of 5- square root of 3)^2

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