Basic Math Examples

$\sqrt{\frac{34}{21}} \div \sqrt[3]{\frac{9}{14}}$

Step 1

Rewrite the division as a fraction.

$\frac{\sqrt{\frac{34}{21}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $\sqrt{\frac{34}{21}}$ as $\frac{\sqrt{34}}{\sqrt{21}}$ .

$\frac{\frac{\sqrt{34}}{\sqrt{21}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.2

Multiply $\frac{\sqrt{34}}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\frac{\frac{\sqrt{34}}{\sqrt{21}} \cdot \frac{\sqrt{21}}{\sqrt{21}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

Multiply $\frac{\sqrt{34}}{\sqrt{21}}$ by $\frac{\sqrt{21}}{\sqrt{21}}$ .

$\frac{\frac{\sqrt{34} \sqrt{21}}{\sqrt{21} \sqrt{21}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3.2

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{\frac{\sqrt{34} \sqrt{21}}{{\sqrt{21}}^{1} \sqrt{21}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3.3

Raise $\sqrt{21}$ to the power of $1$ .

$\frac{\frac{\sqrt{34} \sqrt{21}}{{\sqrt{21}}^{1} {\sqrt{21}}^{1}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3.4

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

$\frac{\frac{\sqrt{34} \sqrt{21}}{{\sqrt{21}}^{1 + 1}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3.5

Add $1$ and $1$ .

$\frac{\frac{\sqrt{34} \sqrt{21}}{{\sqrt{21}}^{2}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3.6

Rewrite ${\sqrt{21}}^{2}$ as $21$ .

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Step 2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{21}$ as $21^{\frac{1}{2}}$ .

$\frac{\frac{\sqrt{34} \sqrt{21}}{{(21^{\frac{1}{2}})}^{2}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3.6.2

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

$\frac{\frac{\sqrt{34} \sqrt{21}}{21^{\frac{1}{2} \cdot 2}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3.6.3

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

$\frac{\frac{\sqrt{34} \sqrt{21}}{21^{\frac{2}{2}}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.6.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{34} \sqrt{21}}{21^{\frac{2}{2}}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{34} \sqrt{21}}{21^{1}}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.3.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{34} \sqrt{21}}{21}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{34 \cdot 21}}{21}}{\sqrt[3]{\frac{9}{14}}}$

Step 2.4.2

Multiply $34$ by $21$ .

$\frac{\frac{\sqrt{714}}{21}}{\sqrt[3]{\frac{9}{14}}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $\sqrt[3]{\frac{9}{14}}$ as $\frac{\sqrt[3]{9}}{\sqrt[3]{14}}$ .

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9}}{\sqrt[3]{14}}}$

Step 3.2

Multiply $\frac{\sqrt[3]{9}}{\sqrt[3]{14}}$ by $\frac{{\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{2}}$ .

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9}}{\sqrt[3]{14}} \cdot \frac{{\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{2}}}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{\sqrt[3]{9}}{\sqrt[3]{14}}$ by $\frac{{\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{2}}$ .

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} {\sqrt[3]{14}}^{2}}{\sqrt[3]{14} {\sqrt[3]{14}}^{2}}}$

Step 3.3.2

Raise $\sqrt[3]{14}$ to the power of $1$ .

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} {\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{1} {\sqrt[3]{14}}^{2}}}$

Step 3.3.3

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

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} {\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{1 + 2}}}$

Step 3.3.4

Add $1$ and $2$ .

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} {\sqrt[3]{14}}^{2}}{{\sqrt[3]{14}}^{3}}}$

Step 3.3.5

Rewrite ${\sqrt[3]{14}}^{3}$ as $14$ .

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Step 3.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{14}$ as $14^{\frac{1}{3}}$ .

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} {\sqrt[3]{14}}^{2}}{{(14^{\frac{1}{3}})}^{3}}}$

Step 3.3.5.2

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

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} {\sqrt[3]{14}}^{2}}{14^{\frac{1}{3} \cdot 3}}}$

Step 3.3.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} {\sqrt[3]{14}}^{2}}{14^{\frac{3}{3}}}}$

Step 3.3.5.4

Cancel the common factor of $3$ .

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Step 3.3.5.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} {\sqrt[3]{14}}^{2}}{14^{\frac{3}{3}}}}$

Step 3.3.5.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} {\sqrt[3]{14}}^{2}}{14^{1}}}$

Step 3.3.5.5

Evaluate the exponent.

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} {\sqrt[3]{14}}^{2}}{14}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Rewrite ${\sqrt[3]{14}}^{2}$ as $\sqrt[3]{14^{2}}$ .

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} \sqrt[3]{14^{2}}}{14}}$

Step 3.4.2

Raise $14$ to the power of $2$ .

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9} \sqrt[3]{196}}{14}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{9 \cdot 196}}{14}}$

Step 3.5.2

Multiply $9$ by $196$ .

$\frac{\frac{\sqrt{714}}{21}}{\frac{\sqrt[3]{1764}}{14}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{714}}{21} \cdot \frac{14}{\sqrt[3]{1764}}$

Step 5

Cancel the common factor of $7$ .

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Step 5.1

Factor $7$ out of $21$ .

$\frac{\sqrt{714}}{7 (3)} \cdot \frac{14}{\sqrt[3]{1764}}$

Step 5.2

Factor $7$ out of $14$ .

$\frac{\sqrt{714}}{7 \cdot 3} \cdot \frac{7 \cdot 2}{\sqrt[3]{1764}}$

Step 5.3

Cancel the common factor.

$\frac{\sqrt{714}}{7 \cdot 3} \cdot \frac{7 \cdot 2}{\sqrt[3]{1764}}$

Step 5.4

Rewrite the expression.

$\frac{\sqrt{714}}{3} \cdot \frac{2}{\sqrt[3]{1764}}$

Step 6

Multiply $\frac{\sqrt{714}}{3}$ by $\frac{2}{\sqrt[3]{1764}}$ .

$\frac{\sqrt{714} \cdot 2}{3 \sqrt[3]{1764}}$

Step 7

Move $2$ to the left of $\sqrt{714}$ .

$\frac{2 \sqrt{714}}{3 \sqrt[3]{1764}}$

Step 8

Multiply $\frac{2 \sqrt{714}}{3 \sqrt[3]{1764}}$ by $\frac{{\sqrt[3]{1764}}^{2}}{{\sqrt[3]{1764}}^{2}}$ .

$\frac{2 \sqrt{714}}{3 \sqrt[3]{1764}} \cdot \frac{{\sqrt[3]{1764}}^{2}}{{\sqrt[3]{1764}}^{2}}$

Step 9

Simplify terms.

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Step 9.1

Combine and simplify the denominator.

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Step 9.1.1

Multiply $\frac{2 \sqrt{714}}{3 \sqrt[3]{1764}}$ by $\frac{{\sqrt[3]{1764}}^{2}}{{\sqrt[3]{1764}}^{2}}$ .

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 \sqrt[3]{1764} {\sqrt[3]{1764}}^{2}}$

Step 9.1.2

Move $\sqrt[3]{1764}$ .

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 (\sqrt[3]{1764} {\sqrt[3]{1764}}^{2})}$

Step 9.1.3

Raise $\sqrt[3]{1764}$ to the power of $1$ .

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 ({\sqrt[3]{1764}}^{1} {\sqrt[3]{1764}}^{2})}$

Step 9.1.4

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

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 {\sqrt[3]{1764}}^{1 + 2}}$

Step 9.1.5

Add $1$ and $2$ .

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 {\sqrt[3]{1764}}^{3}}$

Step 9.1.6

Rewrite ${\sqrt[3]{1764}}^{3}$ as $1764$ .

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Step 9.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{1764}$ as $1764^{\frac{1}{3}}$ .

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 {(1764^{\frac{1}{3}})}^{3}}$

Step 9.1.6.2

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

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 \cdot 1764^{\frac{1}{3} \cdot 3}}$

Step 9.1.6.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 \cdot 1764^{\frac{3}{3}}}$

Step 9.1.6.4

Cancel the common factor of $3$ .

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Step 9.1.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 \cdot 1764^{\frac{3}{3}}}$

Step 9.1.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 \cdot 1764^{1}}$

Step 9.1.6.5

Evaluate the exponent.

$\frac{2 \sqrt{714} {\sqrt[3]{1764}}^{2}}{3 \cdot 1764}$

Step 9.2

Cancel the common factor of $2$ and $1764$ .

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Step 9.2.1

Factor $2$ out of $2 \sqrt{714} {\sqrt[3]{1764}}^{2}$ .

$\frac{2 (\sqrt{714} {\sqrt[3]{1764}}^{2})}{3 \cdot 1764}$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $2$ out of $3 \cdot 1764$ .

$\frac{2 (\sqrt{714} {\sqrt[3]{1764}}^{2})}{2 (3 \cdot 882)}$

Step 9.2.2.2

Cancel the common factor.

$\frac{2 (\sqrt{714} {\sqrt[3]{1764}}^{2})}{2 (3 \cdot 882)}$

Step 9.2.2.3

Rewrite the expression.

$\frac{\sqrt{714} {\sqrt[3]{1764}}^{2}}{3 \cdot 882}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite ${\sqrt[3]{1764}}^{2}$ as $\sqrt[3]{1764^{2}}$ .

$\frac{\sqrt{714} \sqrt[3]{1764^{2}}}{3 \cdot 882}$

Step 10.2

Raise $1764$ to the power of $2$ .

$\frac{\sqrt{714} \sqrt[3]{3111696}}{3 \cdot 882}$

Step 10.3

Rewrite $3111696$ as $42^{3} \cdot 42$ .

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Step 10.3.1

Factor $74088$ out of $3111696$ .

$\frac{\sqrt{714} \sqrt[3]{74088 (42)}}{3 \cdot 882}$

Step 10.3.2

Rewrite $74088$ as $42^{3}$ .

$\frac{\sqrt{714} \sqrt[3]{42^{3} \cdot 42}}{3 \cdot 882}$

Step 10.4

Pull terms out from under the radical.

$\frac{\sqrt{714} \cdot 42 \sqrt[3]{42}}{3 \cdot 882}$

Step 10.5

Combine exponents.

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Step 10.5.1

Rewrite the expression using the least common index of $6$ .

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Step 10.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{714}$ as $714^{\frac{1}{2}}$ .

$\frac{42 (714^{\frac{1}{2}} \sqrt[3]{42})}{3 \cdot 882}$

Step 10.5.1.2

Rewrite $714^{\frac{1}{2}}$ as $714^{\frac{3}{6}}$ .

$\frac{42 (714^{\frac{3}{6}} \sqrt[3]{42})}{3 \cdot 882}$

Step 10.5.1.3

Rewrite $714^{\frac{3}{6}}$ as $\sqrt[6]{714^{3}}$ .

$\frac{42 (\sqrt[6]{714^{3}} \sqrt[3]{42})}{3 \cdot 882}$

Step 10.5.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{42}$ as $42^{\frac{1}{3}}$ .

$\frac{42 (\sqrt[6]{714^{3}} \cdot 42^{\frac{1}{3}})}{3 \cdot 882}$

Step 10.5.1.5

Rewrite $42^{\frac{1}{3}}$ as $42^{\frac{2}{6}}$ .

$\frac{42 (\sqrt[6]{714^{3}} \cdot 42^{\frac{2}{6}})}{3 \cdot 882}$

Step 10.5.1.6

Rewrite $42^{\frac{2}{6}}$ as $\sqrt[6]{42^{2}}$ .

$\frac{42 (\sqrt[6]{714^{3}} \sqrt[6]{42^{2}})}{3 \cdot 882}$

Step 10.5.2

Combine using the product rule for radicals.

$\frac{42 \sqrt[6]{714^{3} \cdot 42^{2}}}{3 \cdot 882}$

Step 10.5.3

Raise $714$ to the power of $3$ .

$\frac{42 \sqrt[6]{363994344 \cdot 42^{2}}}{3 \cdot 882}$

Step 10.5.4

Raise $42$ to the power of $2$ .

$\frac{42 \sqrt[6]{363994344 \cdot 1764}}{3 \cdot 882}$

Step 10.5.5

Multiply $363994344$ by $1764$ .

$\frac{42 \sqrt[6]{642086022816}}{3 \cdot 882}$

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

Multiply $3$ by $882$ .

$\frac{42 \sqrt[6]{642086022816}}{2646}$

Step 11.2

Cancel the common factor of $42$ and $2646$ .

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Step 11.2.1

Factor $42$ out of $42 \sqrt[6]{642086022816}$ .

$\frac{42 (\sqrt[6]{642086022816})}{2646}$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $42$ out of $2646$ .

$\frac{42 \sqrt[6]{642086022816}}{42 \cdot 63}$

Step 11.2.2.2

Cancel the common factor.

$\frac{42 \sqrt[6]{642086022816}}{42 \cdot 63}$

Step 11.2.2.3

Rewrite the expression.

$\frac{\sqrt[6]{642086022816}}{63}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[6]{642086022816}}{63}$

Decimal Form:

$1.47431964 \dots$