Basic Math Examples

$(\sqrt{3 \frac{6}{7}} - \sqrt{1 \frac{5}{7}}) \div \sqrt{\frac{3}{175}}$

Step 1

Convert $3 \frac{6}{7}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$(\sqrt{3 + \frac{6}{7}} - \sqrt{1 \frac{5}{7}}) \div \sqrt{\frac{3}{175}}$

Step 1.2

Add $3$ and $\frac{6}{7}$ .

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Step 1.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$(\sqrt{3 \cdot \frac{7}{7} + \frac{6}{7}} - \sqrt{1 \frac{5}{7}}) \div \sqrt{\frac{3}{175}}$

Step 1.2.2

Combine $3$ and $\frac{7}{7}$ .

$(\sqrt{\frac{3 \cdot 7}{7} + \frac{6}{7}} - \sqrt{1 \frac{5}{7}}) \div \sqrt{\frac{3}{175}}$

Step 1.2.3

Combine the numerators over the common denominator.

$(\sqrt{\frac{3 \cdot 7 + 6}{7}} - \sqrt{1 \frac{5}{7}}) \div \sqrt{\frac{3}{175}}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $3$ by $7$ .

$(\sqrt{\frac{21 + 6}{7}} - \sqrt{1 \frac{5}{7}}) \div \sqrt{\frac{3}{175}}$

Step 1.2.4.2

Add $21$ and $6$ .

$(\sqrt{\frac{27}{7}} - \sqrt{1 \frac{5}{7}}) \div \sqrt{\frac{3}{175}}$

Step 2

Convert $1 \frac{5}{7}$ to an improper fraction.

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Step 2.1

A mixed number is an addition of its whole and fractional parts.

$(\sqrt{\frac{27}{7}} - \sqrt{1 + \frac{5}{7}}) \div \sqrt{\frac{3}{175}}$

Step 2.2

Add $1$ and $\frac{5}{7}$ .

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Step 2.2.1

Write $1$ as a fraction with a common denominator.

$(\sqrt{\frac{27}{7}} - \sqrt{\frac{7}{7} + \frac{5}{7}}) \div \sqrt{\frac{3}{175}}$

Step 2.2.2

Combine the numerators over the common denominator.

$(\sqrt{\frac{27}{7}} - \sqrt{\frac{7 + 5}{7}}) \div \sqrt{\frac{3}{175}}$

Step 2.2.3

Add $7$ and $5$ .

$(\sqrt{\frac{27}{7}} - \sqrt{\frac{12}{7}}) \div \sqrt{\frac{3}{175}}$

Step 3

Rewrite the division as a fraction.

$\frac{\sqrt{\frac{27}{7}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $\sqrt{\frac{27}{7}}$ as $\frac{\sqrt{27}}{\sqrt{7}}$ .

$\frac{\frac{\sqrt{27}}{\sqrt{7}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.2.1.1

Factor $9$ out of $27$ .

$\frac{\frac{\sqrt{9 (3)}}{\sqrt{7}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.2.1.2

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

$\frac{\frac{\sqrt{3^{2} \cdot 3}}{\sqrt{7}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.2.2

Pull terms out from under the radical.

$\frac{\frac{3 \sqrt{3}}{\sqrt{7}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.3

Multiply $\frac{3 \sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\frac{3 \sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{3 \sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4.4

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

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4.5

Add $1$ and $1$ .

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{2}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4.6.2

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

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4.6.3

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

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4.6.4.2

Rewrite the expression.

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{7^{1}} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.4.6.5

Evaluate the exponent.

$\frac{\frac{3 \sqrt{3} \sqrt{7}}{7} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Combine using the product rule for radicals.

$\frac{\frac{3 \sqrt{7 \cdot 3}}{7} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.5.2

Multiply $7$ by $3$ .

$\frac{\frac{3 \sqrt{21}}{7} - \sqrt{\frac{12}{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.6

Rewrite $\sqrt{\frac{12}{7}}$ as $\frac{\sqrt{12}}{\sqrt{7}}$ .

$\frac{\frac{3 \sqrt{21}}{7} - \frac{\sqrt{12}}{\sqrt{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.7.1.1

Factor $4$ out of $12$ .

$\frac{\frac{3 \sqrt{21}}{7} - \frac{\sqrt{4 (3)}}{\sqrt{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.7.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\frac{3 \sqrt{21}}{7} - \frac{\sqrt{2^{2} \cdot 3}}{\sqrt{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.7.2

Pull terms out from under the radical.

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3}}{\sqrt{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.8

Multiply $\frac{2 \sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\frac{3 \sqrt{21}}{7} - (\frac{2 \sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})}{\sqrt{\frac{3}{175}}}$

Step 4.9

Combine and simplify the denominator.

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Step 4.9.1

Multiply $\frac{2 \sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.9.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}}{\sqrt{\frac{3}{175}}}$

Step 4.9.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}}{\sqrt{\frac{3}{175}}}$

Step 4.9.4

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

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}}{\sqrt{\frac{3}{175}}}$

Step 4.9.5

Add $1$ and $1$ .

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{{\sqrt{7}}^{2}}}{\sqrt{\frac{3}{175}}}$

Step 4.9.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}}{\sqrt{\frac{3}{175}}}$

Step 4.9.6.2

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

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}}{\sqrt{\frac{3}{175}}}$

Step 4.9.6.3

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

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}}{\sqrt{\frac{3}{175}}}$

Step 4.9.6.4

Cancel the common factor of $2$ .

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Step 4.9.6.4.1

Cancel the common factor.

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}}{\sqrt{\frac{3}{175}}}$

Step 4.9.6.4.2

Rewrite the expression.

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{7^{1}}}{\sqrt{\frac{3}{175}}}$

Step 4.9.6.5

Evaluate the exponent.

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{3} \sqrt{7}}{7}}{\sqrt{\frac{3}{175}}}$

Step 4.10

Simplify the numerator.

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Step 4.10.1

Combine using the product rule for radicals.

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{7 \cdot 3}}{7}}{\sqrt{\frac{3}{175}}}$

Step 4.10.2

Multiply $7$ by $3$ .

$\frac{\frac{3 \sqrt{21}}{7} - \frac{2 \sqrt{21}}{7}}{\sqrt{\frac{3}{175}}}$

Step 4.11

Combine the numerators over the common denominator.

$\frac{\frac{3 \sqrt{21} - 2 \sqrt{21}}{7}}{\sqrt{\frac{3}{175}}}$

Step 4.12

Subtract $2 \sqrt{21}$ from $3 \sqrt{21}$ .

$\frac{\frac{\sqrt{21}}{7}}{\sqrt{\frac{3}{175}}}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $\sqrt{\frac{3}{175}}$ as $\frac{\sqrt{3}}{\sqrt{175}}$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3}}{\sqrt{175}}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Rewrite $175$ as $5^{2} \cdot 7$ .

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Step 5.2.1.1

Factor $25$ out of $175$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3}}{\sqrt{25 (7)}}}$

Step 5.2.1.2

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

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3}}{\sqrt{5^{2} \cdot 7}}}$

Step 5.2.2

Pull terms out from under the radical.

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3}}{5 \sqrt{7}}}$

Step 5.3

Multiply $\frac{\sqrt{3}}{5 \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3}}{5 \sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}}$

Step 5.4

Combine and simplify the denominator.

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Step 5.4.1

Multiply $\frac{\sqrt{3}}{5 \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 \sqrt{7} \sqrt{7}}}$

Step 5.4.2

Move $\sqrt{7}$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 (\sqrt{7} \sqrt{7})}}$

Step 5.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 ({\sqrt{7}}^{1} \sqrt{7})}}$

Step 5.4.4

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}}$

Step 5.4.5

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

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 {\sqrt{7}}^{1 + 1}}}$

Step 5.4.6

Add $1$ and $1$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 {\sqrt{7}}^{2}}}$

Step 5.4.7

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 5.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 {(7^{\frac{1}{2}})}^{2}}}$

Step 5.4.7.2

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

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 \cdot 7^{\frac{1}{2} \cdot 2}}}$

Step 5.4.7.3

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

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 \cdot 7^{\frac{2}{2}}}}$

Step 5.4.7.4

Cancel the common factor of $2$ .

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Step 5.4.7.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 \cdot 7^{\frac{2}{2}}}}$

Step 5.4.7.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 \cdot 7^{1}}}$

Step 5.4.7.5

Evaluate the exponent.

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3} \sqrt{7}}{5 \cdot 7}}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{3 \cdot 7}}{5 \cdot 7}}$

Step 5.5.2

Multiply $3$ by $7$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{21}}{5 \cdot 7}}$

Step 5.6

Multiply $5$ by $7$ .

$\frac{\frac{\sqrt{21}}{7}}{\frac{\sqrt{21}}{35}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{21}}{7} \cdot \frac{35}{\sqrt{21}}$

Step 7

Cancel the common factor of $\sqrt{21}$ .

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Step 7.1

Cancel the common factor.

$\frac{\sqrt{21}}{7} \cdot \frac{35}{\sqrt{21}}$

Step 7.2

Rewrite the expression.

$\frac{1}{7} \cdot 35$

Step 8

Cancel the common factor of $7$ .

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Step 8.1

Factor $7$ out of $35$ .

$\frac{1}{7} \cdot (7 (5))$

Step 8.2

Cancel the common factor.

$\frac{1}{7} \cdot (7 \cdot 5)$

Step 8.3

Rewrite the expression.

$5$