Basic Math Examples

${({(\sqrt{a})}^{3} \cdot \sqrt[4]{a^{5}})}^{2} \div \sqrt{a^{7}}$

Step 1

Rewrite the division as a fraction.

$\frac{{({(\sqrt{a})}^{3} \cdot \sqrt[4]{a^{5}})}^{2}}{\sqrt{a^{7}}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to ${\sqrt{a}}^{3} \cdot \sqrt[4]{a^{5}}$ .

$\frac{{({\sqrt{a}}^{3})}^{2} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.2

Multiply the exponents in ${({\sqrt{a}}^{3})}^{2}$ .

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Step 2.2.1

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

$\frac{{\sqrt{a}}^{3 \cdot 2} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.2.2

Multiply $3$ by $2$ .

$\frac{{\sqrt{a}}^{6} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.3

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

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Step 2.3.1

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

$\frac{{(a^{\frac{1}{2}})}^{6} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.3.2

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

$\frac{a^{\frac{1}{2} \cdot 6} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.3.3

Combine $\frac{1}{2}$ and $6$ .

$\frac{a^{\frac{6}{2}} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.3.4

Cancel the common factor of $6$ and $2$ .

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Step 2.3.4.1

Factor $2$ out of $6$ .

$\frac{a^{\frac{2 \cdot 3}{2}} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.3.4.2

Cancel the common factors.

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Step 2.3.4.2.1

Factor $2$ out of $2$ .

$\frac{a^{\frac{2 \cdot 3}{2 (1)}} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.3.4.2.2

Cancel the common factor.

$\frac{a^{\frac{2 \cdot 3}{2 \cdot 1}} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.3.4.2.3

Rewrite the expression.

$\frac{a^{\frac{3}{1}} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.3.4.2.4

Divide $3$ by $1$ .

$\frac{a^{3} \cdot {\sqrt[4]{a^{5}}}^{2}}{\sqrt{a^{7}}}$

Step 2.4

Factor out $a^{4}$ .

$\frac{a^{3} \cdot {\sqrt[4]{a^{4} a}}^{2}}{\sqrt{a^{7}}}$

Step 2.5

Pull terms out from under the radical.

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

Step 2.6

Apply the product rule to $a \sqrt[4]{a}$ .

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

Step 2.7

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

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Step 2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{a}$ as $a^{\frac{1}{4}}$ .

$\frac{a^{3} \cdot (a^{2} {(a^{\frac{1}{4}})}^{2})}{\sqrt{a^{7}}}$

Step 2.7.2

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

$\frac{a^{3} \cdot (a^{2} a^{\frac{1}{4} \cdot 2})}{\sqrt{a^{7}}}$

Step 2.7.3

Combine $\frac{1}{4}$ and $2$ .

$\frac{a^{3} \cdot (a^{2} a^{\frac{2}{4}})}{\sqrt{a^{7}}}$

Step 2.7.4

Cancel the common factor of $2$ and $4$ .

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Step 2.7.4.1

Factor $2$ out of $2$ .

$\frac{a^{3} \cdot (a^{2} a^{\frac{2 (1)}{4}})}{\sqrt{a^{7}}}$

Step 2.7.4.2

Cancel the common factors.

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Step 2.7.4.2.1

Factor $2$ out of $4$ .

$\frac{a^{3} \cdot (a^{2} a^{\frac{2 \cdot 1}{2 \cdot 2}})}{\sqrt{a^{7}}}$

Step 2.7.4.2.2

Cancel the common factor.

$\frac{a^{3} \cdot (a^{2} a^{\frac{2 \cdot 1}{2 \cdot 2}})}{\sqrt{a^{7}}}$

Step 2.7.4.2.3

Rewrite the expression.

$\frac{a^{3} \cdot (a^{2} a^{\frac{1}{2}})}{\sqrt{a^{7}}}$

Step 2.7.5

Rewrite $a^{\frac{1}{2}}$ as $\sqrt{a}$ .

$\frac{a^{3} \cdot (a^{2} \sqrt{a})}{\sqrt{a^{7}}}$

$\frac{a^{3} \cdot a^{2} \sqrt{a}}{\sqrt{a^{7}}}$

Step 2.8

Multiply $a^{3}$ by $a^{2}$ by adding the exponents.

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Step 2.8.1

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

$\frac{a^{3 + 2} \sqrt{a}}{\sqrt{a^{7}}}$

Step 2.8.2

Add $3$ and $2$ .

$\frac{a^{5} \sqrt{a}}{\sqrt{a^{7}}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $a^{7}$ as ${(a^{3})}^{2} a$ .

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Step 3.1.1

Factor out $a^{6}$ .

$\frac{a^{5} \sqrt{a}}{\sqrt{a^{6} a}}$

Step 3.1.2

Rewrite $a^{6}$ as ${(a^{3})}^{2}$ .

$\frac{a^{5} \sqrt{a}}{\sqrt{{(a^{3})}^{2} a}}$

Step 3.2

Pull terms out from under the radical.

$\frac{a^{5} \sqrt{a}}{a^{3} \sqrt{a}}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $a^{5}$ and $a^{3}$ .

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Step 4.1.1

Factor $a^{3}$ out of $a^{5} \sqrt{a}$ .

$\frac{a^{3} (a^{2} \sqrt{a})}{a^{3} \sqrt{a}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $a^{3}$ out of $a^{3} \sqrt{a}$ .

$\frac{a^{3} (a^{2} \sqrt{a})}{a^{3} (\sqrt{a})}$

Step 4.1.2.2

Cancel the common factor.

$\frac{a^{3} (a^{2} \sqrt{a})}{a^{3} \sqrt{a}}$

Step 4.1.2.3

Rewrite the expression.

$\frac{a^{2} \sqrt{a}}{\sqrt{a}}$

Step 4.2

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

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Step 4.2.1

Cancel the common factor.

$\frac{a^{2} \sqrt{a}}{\sqrt{a}}$

Step 4.2.2

Divide $a^{2}$ by $1$ .

$a^{2}$