Basic Math Examples

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

Step 1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Divide $\sqrt{2}$ by $1$ .

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

Step 4

Rewrite $\sqrt{\sqrt{2}}$ as $\sqrt[4]{2}$ .

$\sqrt{\frac{5}{\sqrt{5} + 3} - \sqrt[4]{2}}$

Step 5

To write $- \sqrt[4]{2}$ as a fraction with a common denominator, multiply by $\frac{\sqrt{5} + 3}{\sqrt{5} + 3}$ .

$\sqrt{\frac{5}{\sqrt{5} + 3} - \sqrt[4]{2} \cdot \frac{\sqrt{5} + 3}{\sqrt{5} + 3}}$

Step 6

Combine $- \sqrt[4]{2}$ and $\frac{\sqrt{5} + 3}{\sqrt{5} + 3}$ .

$\sqrt{\frac{5}{\sqrt{5} + 3} + \frac{- \sqrt[4]{2} (\sqrt{5} + 3)}{\sqrt{5} + 3}}$

Step 7

Combine the numerators over the common denominator.

$\sqrt{\frac{5 - \sqrt[4]{2} (\sqrt{5} + 3)}{\sqrt{5} + 3}}$

Step 8

Rewrite $\frac{5 - \sqrt[4]{2} (\sqrt{5} + 3)}{\sqrt{5} + 3}$ in a factored form.

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Step 8.1

Apply the distributive property.

$\sqrt{\frac{5 - \sqrt[4]{2} \sqrt{5} - \sqrt[4]{2} \cdot 3}{\sqrt{5} + 3}}$

Step 8.2

Multiply $- \sqrt[4]{2} \sqrt{5}$ .

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Step 8.2.1

Rewrite the expression using the least common index of $4$ .

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Step 8.2.1.1

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

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

Step 8.2.1.2

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{2}{4}}$ .

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

Step 8.2.1.3

Rewrite $5^{\frac{2}{4}}$ as $\sqrt[4]{5^{2}}$ .

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

Step 8.2.2

Combine using the product rule for radicals.

$\sqrt{\frac{5 - \sqrt[4]{5^{2} \cdot 2} - \sqrt[4]{2} \cdot 3}{\sqrt{5} + 3}}$

Step 8.2.3

Raise $5$ to the power of $2$ .

$\sqrt{\frac{5 - \sqrt[4]{25 \cdot 2} - \sqrt[4]{2} \cdot 3}{\sqrt{5} + 3}}$

Step 8.2.4

Multiply $25$ by $2$ .

$\sqrt{\frac{5 - \sqrt[4]{50} - \sqrt[4]{2} \cdot 3}{\sqrt{5} + 3}}$

Step 8.3

Multiply $3$ by $- 1$ .

$\sqrt{\frac{5 - \sqrt[4]{50} - 3 \sqrt[4]{2}}{\sqrt{5} + 3}}$

Step 9

Multiply $\frac{5 - \sqrt[4]{50} - 3 \sqrt[4]{2}}{\sqrt{5} + 3}$ by $\frac{\sqrt{5} - 3}{\sqrt{5} - 3}$ .

$\sqrt{\frac{5 - \sqrt[4]{50} - 3 \sqrt[4]{2}}{\sqrt{5} + 3} \cdot \frac{\sqrt{5} - 3}{\sqrt{5} - 3}}$

Step 10

Combine fractions.

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Step 10.1

Multiply $\frac{5 - \sqrt[4]{50} - 3 \sqrt[4]{2}}{\sqrt{5} + 3}$ by $\frac{\sqrt{5} - 3}{\sqrt{5} - 3}$ .

$\sqrt{\frac{(5 - \sqrt[4]{50} - 3 \sqrt[4]{2}) (\sqrt{5} - 3)}{(\sqrt{5} + 3) (\sqrt{5} - 3)}}$

Step 10.2

Expand the denominator using the FOIL method.

$\sqrt{\frac{(5 - \sqrt[4]{50} - 3 \sqrt[4]{2}) (\sqrt{5} - 3)}{{\sqrt{5}}^{2} + \sqrt{5} \cdot - 3 + 3 \sqrt{5} - 9}}$

Step 10.3

Simplify.

$\sqrt{\frac{(5 - \sqrt[4]{50} - 3 \sqrt[4]{2}) (\sqrt{5} - 3)}{- 4}}$

Step 10.4

Move the negative in front of the fraction.

$\sqrt{- \frac{(5 - \sqrt[4]{50} - 3 \sqrt[4]{2}) (\sqrt{5} - 3)}{4}}$

Step 11

Expand $(5 - \sqrt[4]{50} - 3 \sqrt[4]{2}) (\sqrt{5} - 3)$ by multiplying each term in the first expression by each term in the second expression.

$\sqrt{- \frac{5 \sqrt{5} + 5 \cdot - 3 - \sqrt[4]{50} \sqrt{5} - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 12

Multiply $5$ by $- 3$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - \sqrt[4]{50} \sqrt{5} - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 13

Multiply $- \sqrt[4]{50} \sqrt{5}$ .

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Step 13.1

Rewrite the expression using the least common index of $4$ .

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Step 13.1.1

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

$\sqrt{- \frac{5 \sqrt{5} - 15 - (5^{\frac{1}{2}} \sqrt[4]{50}) - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 13.1.2

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{2}{4}}$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - (5^{\frac{2}{4}} \sqrt[4]{50}) - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 13.1.3

Rewrite $5^{\frac{2}{4}}$ as $\sqrt[4]{5^{2}}$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - (\sqrt[4]{5^{2}} \sqrt[4]{50}) - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 13.2

Combine using the product rule for radicals.

$\sqrt{- \frac{5 \sqrt{5} - 15 - \sqrt[4]{5^{2} \cdot 50} - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 13.3

Raise $5$ to the power of $2$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - \sqrt[4]{25 \cdot 50} - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 13.4

Multiply $25$ by $50$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - \sqrt[4]{1250} - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 14

Rewrite $1250$ as $5^{4} \cdot 2$ .

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Step 14.1

Factor $625$ out of $1250$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - \sqrt[4]{625 (2)} - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 14.2

Rewrite $625$ as $5^{4}$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - \sqrt[4]{5^{4} \cdot 2} - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 15

Pull terms out from under the radical.

$\sqrt{- \frac{5 \sqrt{5} - 15 - (5 \sqrt[4]{2}) - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 16

Multiply.

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Step 16.1

Multiply $5$ by $- 1$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - 5 \sqrt[4]{2} - \sqrt[4]{50} \cdot - 3 - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 16.2

Multiply $- 3$ by $- 1$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - 5 \sqrt[4]{2} + 3 \sqrt[4]{50} - 3 \sqrt[4]{2} \sqrt{5} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 17

Multiply $- 3 \sqrt[4]{2} \sqrt{5}$ .

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Step 17.1

Rewrite the expression using the least common index of $4$ .

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Step 17.1.1

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

$\sqrt{- \frac{5 \sqrt{5} - 15 - 5 \sqrt[4]{2} + 3 \sqrt[4]{50} - 3 (5^{\frac{1}{2}} \sqrt[4]{2}) - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 17.1.2

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{2}{4}}$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - 5 \sqrt[4]{2} + 3 \sqrt[4]{50} - 3 (5^{\frac{2}{4}} \sqrt[4]{2}) - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 17.1.3

Rewrite $5^{\frac{2}{4}}$ as $\sqrt[4]{5^{2}}$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - 5 \sqrt[4]{2} + 3 \sqrt[4]{50} - 3 (\sqrt[4]{5^{2}} \sqrt[4]{2}) - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 17.2

Combine using the product rule for radicals.

$\sqrt{- \frac{5 \sqrt{5} - 15 - 5 \sqrt[4]{2} + 3 \sqrt[4]{50} - 3 \sqrt[4]{5^{2} \cdot 2} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 17.3

Raise $5$ to the power of $2$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - 5 \sqrt[4]{2} + 3 \sqrt[4]{50} - 3 \sqrt[4]{25 \cdot 2} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 17.4

Multiply $25$ by $2$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - 5 \sqrt[4]{2} + 3 \sqrt[4]{50} - 3 \sqrt[4]{50} - 3 \sqrt[4]{2} \cdot - 3}{4}}$

Step 18

Multiply $- 3$ by $- 3$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 - 5 \sqrt[4]{2} + 3 \sqrt[4]{50} - 3 \sqrt[4]{50} + 9 \sqrt[4]{2}}{4}}$

Step 19

Simplify the numerator.

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Step 19.1

Add $- 5 \sqrt[4]{2}$ and $9 \sqrt[4]{2}$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 + 4 \sqrt[4]{2} + 3 \sqrt[4]{50} - 3 \sqrt[4]{50}}{4}}$

Step 19.2

Subtract $3 \sqrt[4]{50}$ from $3 \sqrt[4]{50}$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 + 4 \sqrt[4]{2} + 0}{4}}$

Step 19.3

Add $5 \sqrt{5} - 15 + 4 \sqrt[4]{2}$ and $0$ .

$\sqrt{- \frac{5 \sqrt{5} - 15 + 4 \sqrt[4]{2}}{4}}$

Step 20

Rewrite $- \frac{5 \sqrt{5} - 15 + 4 \sqrt[4]{2}}{4}$ as ${(\frac{i}{2})}^{2} (5 \sqrt{5} - 15 + 4 \sqrt[4]{2})$ .

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Step 20.1

Rewrite $- 1$ as $i^{2}$ .

$\sqrt{i^{2} \frac{5 \sqrt{5} - 15 + 4 \sqrt[4]{2}}{4}}$

Step 20.2

Factor the perfect power $1^{2}$ out of $5 \sqrt{5} - 15 + 4 \sqrt[4]{2}$ .

$\sqrt{i^{2} \frac{1^{2} (5 \sqrt{5} - 15 + 4 \sqrt[4]{2})}{4}}$

Step 20.3

Factor the perfect power $2^{2}$ out of $4$ .

$\sqrt{i^{2} \frac{1^{2} (5 \sqrt{5} - 15 + 4 \sqrt[4]{2})}{2^{2} \cdot 1}}$

Step 20.4

Rearrange the fraction $\frac{1^{2} (5 \sqrt{5} - 15 + 4 \sqrt[4]{2})}{2^{2} \cdot 1}$ .

$\sqrt{i^{2} ({(\frac{1}{2})}^{2} (5 \sqrt{5} - 15 + 4 \sqrt[4]{2}))}$

Step 20.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

$\sqrt{{(\frac{i}{2})}^{2} (5 \sqrt{5} - 15 + 4 \sqrt[4]{2})}$

Step 21

Pull terms out from under the radical.

$\frac{i}{2} \sqrt{5 \sqrt{5} - 15 + 4 \sqrt[4]{2}}$

Step 22

Combine $\frac{i}{2}$ and $\sqrt{5 \sqrt{5} - 15 + 4 \sqrt[4]{2}}$ .

$\frac{i \sqrt{5 \sqrt{5} - 15 + 4 \sqrt[4]{2}}}{2}$