Basic Math Examples

${(\sqrt{\sqrt[3]{a}} \cdot \sqrt[3]{a^{5}})}^{7} \div {(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}$

Step 1

Rewrite the division as a fraction.

$\frac{{(\sqrt{\sqrt[3]{a}} \cdot \sqrt[3]{a^{5}})}^{7}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $\sqrt{\sqrt[3]{a}} \cdot \sqrt[3]{a^{5}}$ .

$\frac{{\sqrt{\sqrt[3]{a}}}^{7} \cdot {\sqrt[3]{a^{5}}}^{7}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.2

Rewrite $\sqrt{\sqrt[3]{a}}$ as $\sqrt[6]{a}$ .

$\frac{{\sqrt[6]{a}}^{7} \cdot {\sqrt[3]{a^{5}}}^{7}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.3

Rewrite ${\sqrt[6]{a}}^{7}$ as $\sqrt[6]{a^{7}}$ .

$\frac{\sqrt[6]{a^{7}} \cdot {\sqrt[3]{a^{5}}}^{7}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.4

Factor out $a^{6}$ .

$\frac{\sqrt[6]{a^{6} a} \cdot {\sqrt[3]{a^{5}}}^{7}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.5

Pull terms out from under the radical.

$\frac{a \sqrt[6]{a} \cdot {\sqrt[3]{a^{5}}}^{7}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.6

Factor out $a^{3}$ .

$\frac{a \sqrt[6]{a} \cdot {\sqrt[3]{a^{3} a^{2}}}^{7}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.7

Pull terms out from under the radical.

$\frac{a \sqrt[6]{a} \cdot {(a \sqrt[3]{a^{2}})}^{7}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.8

Apply the product rule to $a \sqrt[3]{a^{2}}$ .

$\frac{a \sqrt[6]{a} \cdot (a^{7} {\sqrt[3]{a^{2}}}^{7})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.9

Rewrite ${\sqrt[3]{a^{2}}}^{7}$ as $\sqrt[3]{{(a^{2})}^{7}}$ .

$\frac{a \sqrt[6]{a} \cdot (a^{7} \sqrt[3]{{(a^{2})}^{7}})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.10

Multiply the exponents in ${(a^{2})}^{7}$ .

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Step 2.10.1

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

$\frac{a \sqrt[6]{a} \cdot (a^{7} \sqrt[3]{a^{2 \cdot 7}})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.10.2

Multiply $2$ by $7$ .

$\frac{a \sqrt[6]{a} \cdot (a^{7} \sqrt[3]{a^{14}})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.11

Rewrite $a^{14}$ as ${(a^{4})}^{3} a^{2}$ .

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Step 2.11.1

Factor out $a^{12}$ .

$\frac{a \sqrt[6]{a} \cdot (a^{7} \sqrt[3]{a^{12} a^{2}})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.11.2

Rewrite $a^{12}$ as ${(a^{4})}^{3}$ .

$\frac{a \sqrt[6]{a} \cdot (a^{7} \sqrt[3]{{(a^{4})}^{3} a^{2}})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.12

Pull terms out from under the radical.

$\frac{a \sqrt[6]{a} \cdot (a^{7} (a^{4} \sqrt[3]{a^{2}}))}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.13

Multiply $a^{7}$ by $a^{4}$ by adding the exponents.

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Step 2.13.1

Move $a^{4}$ .

$\frac{a \sqrt[6]{a} \cdot (a^{4} a^{7} \sqrt[3]{a^{2}})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.13.2

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

$\frac{a \sqrt[6]{a} \cdot (a^{4 + 7} \sqrt[3]{a^{2}})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.13.3

Add $4$ and $7$ .

$\frac{a \sqrt[6]{a} \cdot (a^{11} \sqrt[3]{a^{2}})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

$\frac{a \sqrt[6]{a} \cdot a^{11} \sqrt[3]{a^{2}}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14

Combine exponents.

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Step 2.14.1

Multiply $a$ by $a^{11}$ by adding the exponents.

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Step 2.14.1.1

Move $a^{11}$ .

$\frac{a^{11} a \sqrt[6]{a} \sqrt[3]{a^{2}}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14.1.2

Multiply $a^{11}$ by $a$ .

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Step 2.14.1.2.1

Raise $a$ to the power of $1$ .

$\frac{a^{11} a^{1} \sqrt[6]{a} \sqrt[3]{a^{2}}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14.1.2.2

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

$\frac{a^{11 + 1} \sqrt[6]{a} \sqrt[3]{a^{2}}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14.1.3

Add $11$ and $1$ .

$\frac{a^{12} \sqrt[6]{a} \sqrt[3]{a^{2}}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14.2

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

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Step 2.14.2.1

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

$\frac{a^{12} (a^{\frac{2}{3}} \sqrt[6]{a})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14.2.2

Rewrite $a^{\frac{2}{3}}$ as $a^{\frac{4}{6}}$ .

$\frac{a^{12} (a^{\frac{4}{6}} \sqrt[6]{a})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14.2.3

Rewrite $a^{\frac{4}{6}}$ as $\sqrt[6]{a^{4}}$ .

$\frac{a^{12} (\sqrt[6]{a^{4}} \sqrt[6]{a})}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14.3

Combine using the product rule for radicals.

$\frac{a^{12} \sqrt[6]{a^{4} a}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14.4

Multiply $a^{4}$ by $a$ by adding the exponents.

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Step 2.14.4.1

Multiply $a^{4}$ by $a$ .

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Step 2.14.4.1.1

Raise $a$ to the power of $1$ .

$\frac{a^{12} \sqrt[6]{a^{4} a^{1}}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14.4.1.2

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

$\frac{a^{12} \sqrt[6]{a^{4 + 1}}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 2.14.4.2

Add $4$ and $1$ .

$\frac{a^{12} \sqrt[6]{a^{5}}}{{(\sqrt[3]{a^{2}} \div \sqrt{a})}^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $\sqrt[3]{a^{2}} \div \sqrt{a}$ .

$\frac{a^{12} \sqrt[6]{a^{5}}}{{\sqrt[3]{a^{2}}}^{2} \div {\sqrt{a}}^{2}}$

Step 3.2

Rewrite the division as a fraction.

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{{\sqrt[3]{a^{2}}}^{2}}{{\sqrt{a}}^{2}}}$

Step 3.3

Simplify the numerator.

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Step 3.3.1

Rewrite ${\sqrt[3]{a^{2}}}^{2}$ as $\sqrt[3]{{(a^{2})}^{2}}$ .

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{\sqrt[3]{{(a^{2})}^{2}}}{{\sqrt{a}}^{2}}}$

Step 3.3.2

Multiply the exponents in ${(a^{2})}^{2}$ .

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Step 3.3.2.1

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

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{\sqrt[3]{a^{2 \cdot 2}}}{{\sqrt{a}}^{2}}}$

Step 3.3.2.2

Multiply $2$ by $2$ .

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{\sqrt[3]{a^{4}}}{{\sqrt{a}}^{2}}}$

Step 3.3.3

Factor out $a^{3}$ .

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{\sqrt[3]{a^{3} a}}{{\sqrt{a}}^{2}}}$

Step 3.3.4

Pull terms out from under the radical.

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{a \sqrt[3]{a}}{{\sqrt{a}}^{2}}}$

Step 3.4

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

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Step 3.4.1

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

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{a \sqrt[3]{a}}{{(a^{\frac{1}{2}})}^{2}}}$

Step 3.4.2

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

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{a \sqrt[3]{a}}{a^{\frac{1}{2} \cdot 2}}}$

Step 3.4.3

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

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{a \sqrt[3]{a}}{a^{\frac{2}{2}}}}$

Step 3.4.4

Cancel the common factor of $2$ .

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Step 3.4.4.1

Cancel the common factor.

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{a \sqrt[3]{a}}{a^{\frac{2}{2}}}}$

Step 3.4.4.2

Rewrite the expression.

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{a \sqrt[3]{a}}{a^{1}}}$

Step 3.4.5

Simplify.

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{a \sqrt[3]{a}}{a}}$

Step 3.5

Cancel the common factor of $a$ .

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Step 3.5.1

Cancel the common factor.

$\frac{a^{12} \sqrt[6]{a^{5}}}{\frac{a \sqrt[3]{a}}{a}}$

Step 3.5.2

Divide $\sqrt[3]{a}$ by $1$ .

$\frac{a^{12} \sqrt[6]{a^{5}}}{\sqrt[3]{a}}$

Step 4

Multiply $\frac{a^{12} \sqrt[6]{a^{5}}}{\sqrt[3]{a}}$ by $\frac{{\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{2}}$ .

$\frac{a^{12} \sqrt[6]{a^{5}}}{\sqrt[3]{a}} \cdot \frac{{\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{2}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{a^{12} \sqrt[6]{a^{5}}}{\sqrt[3]{a}}$ by $\frac{{\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{2}}$ .

$\frac{a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{\sqrt[3]{a} {\sqrt[3]{a}}^{2}}$

Step 5.2

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

$\frac{a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{1} {\sqrt[3]{a}}^{2}}$

Step 5.3

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

$\frac{a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{1 + 2}}$

Step 5.4

Add $1$ and $2$ .

$\frac{a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{{\sqrt[3]{a}}^{3}}$

Step 5.5

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

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Step 5.5.1

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

$\frac{a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{{(a^{\frac{1}{3}})}^{3}}$

Step 5.5.2

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

$\frac{a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{a^{\frac{1}{3} \cdot 3}}$

Step 5.5.3

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

$\frac{a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{a^{\frac{3}{3}}}$

Step 5.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.1

Cancel the common factor.

$\frac{a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{a^{\frac{3}{3}}}$

Step 5.5.4.2

Rewrite the expression.

$\frac{a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{a^{1}}$

Step 5.5.5

Simplify.

$\frac{a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{a}$

Step 6

Cancel the common factor of $a^{12}$ and $a$ .

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Step 6.1

Factor $a$ out of $a^{12} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}$ .

$\frac{a (a^{11} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2})}{a}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Raise $a$ to the power of $1$ .

$\frac{a (a^{11} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2})}{a^{1}}$

Step 6.2.2

Factor $a$ out of $a^{1}$ .

$\frac{a (a^{11} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2})}{a \cdot 1}$

Step 6.2.3

Cancel the common factor.

$\frac{a (a^{11} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2})}{a \cdot 1}$

Step 6.2.4

Rewrite the expression.

$\frac{a^{11} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}}{1}$

Step 6.2.5

Divide $a^{11} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}$ by $1$ .

$a^{11} \sqrt[6]{a^{5}} {\sqrt[3]{a}}^{2}$

Step 7

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

$a^{11} \sqrt[6]{a^{5}} \sqrt[3]{a^{2}}$

Step 8

Multiply $a^{11} \sqrt[6]{a^{5}} \sqrt[3]{a^{2}}$ .

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Step 8.1

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

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Step 8.1.1

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

$a^{11} (a^{\frac{2}{3}} \sqrt[6]{a^{5}})$

Step 8.1.2

Rewrite $a^{\frac{2}{3}}$ as $a^{\frac{4}{6}}$ .

$a^{11} (a^{\frac{4}{6}} \sqrt[6]{a^{5}})$

Step 8.1.3

Rewrite $a^{\frac{4}{6}}$ as $\sqrt[6]{a^{4}}$ .

$a^{11} (\sqrt[6]{a^{4}} \sqrt[6]{a^{5}})$

Step 8.2

Combine using the product rule for radicals.

$a^{11} \sqrt[6]{a^{4} a^{5}}$

Step 8.3

Multiply $a^{4}$ by $a^{5}$ by adding the exponents.

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Step 8.3.1

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

$a^{11} \sqrt[6]{a^{4 + 5}}$

Step 8.3.2

Add $4$ and $5$ .

$a^{11} \sqrt[6]{a^{9}}$

Step 9

Factor out $a^{6}$ .

$a^{11} \sqrt[6]{a^{6} a^{3}}$

Step 10

Pull terms out from under the radical.

$a^{11} (a \sqrt[6]{a^{3}})$

Step 11

Multiply $a^{11}$ by $a$ by adding the exponents.

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Step 11.1

Move $a$ .

$a \cdot a^{11} \sqrt[6]{a^{3}}$

Step 11.2

Multiply $a$ by $a^{11}$ .

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Step 11.2.1

Raise $a$ to the power of $1$ .

$a^{1} a^{11} \sqrt[6]{a^{3}}$

Step 11.2.2

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

$a^{1 + 11} \sqrt[6]{a^{3}}$

Step 11.3

Add $1$ and $11$ .

$a^{12} \sqrt[6]{a^{3}}$

Step 12

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

$a^{12} \sqrt{\sqrt[3]{a^{3}}}$

Step 13

Pull terms out from under the radical, assuming real numbers.

$a^{12} \sqrt{a}$