Algebra Examples

$40 x^{2} \sqrt{x^{12}} \div 10 x^{2} \sqrt{29 x^{8}}$

Step 1

Rewrite the division as a fraction.

$\frac{40 x^{2} \sqrt{x^{12}}}{10 x^{2} \sqrt{29 x^{8}}}$

Step 2

Cancel the common factor of $40$ and $10$ .

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Step 2.1

Factor $10$ out of $40 x^{2} \sqrt{x^{12}}$ .

$\frac{10 (4 x^{2} \sqrt{x^{12}})}{10 x^{2} \sqrt{29 x^{8}}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $10$ out of $10 x^{2} \sqrt{29 x^{8}}$ .

$\frac{10 (4 x^{2} \sqrt{x^{12}})}{10 (x^{2} \sqrt{29 x^{8}})}$

Step 2.2.2

Cancel the common factor.

$\frac{10 (4 x^{2} \sqrt{x^{12}})}{10 (x^{2} \sqrt{29 x^{8}})}$

Step 2.2.3

Rewrite the expression.

$\frac{4 x^{2} \sqrt{x^{12}}}{x^{2} \sqrt{29 x^{8}}}$

Step 3

Cancel the common factor of $x^{2}$ .

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Step 3.1

Cancel the common factor.

$\frac{4 x^{2} \sqrt{x^{12}}}{x^{2} \sqrt{29 x^{8}}}$

Step 3.2

Rewrite the expression.

$\frac{4 \sqrt{x^{12}}}{\sqrt{29 x^{8}}}$

Step 4

Combine $\sqrt{x^{12}}$ and $\sqrt{29 x^{8}}$ into a single radical.

$4 \sqrt{\frac{x^{12}}{29 x^{8}}}$

Step 5

Reduce the expression $\frac{x^{12}}{29 x^{8}}$ by cancelling the common factors.

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Step 5.1

Factor $x^{8}$ out of $x^{12}$ .

$4 \sqrt{\frac{x^{8} x^{4}}{29 x^{8}}}$

Step 5.2

Factor $x^{8}$ out of $29 x^{8}$ .

$4 \sqrt{\frac{x^{8} x^{4}}{x^{8} \cdot 29}}$

Step 5.3

Cancel the common factor.

$4 \sqrt{\frac{x^{8} x^{4}}{x^{8} \cdot 29}}$

Step 5.4

Rewrite the expression.

$4 \sqrt{\frac{x^{4}}{29}}$

Step 6

Rewrite $\sqrt{\frac{x^{4}}{29}}$ as $\frac{\sqrt{x^{4}}}{\sqrt{29}}$ .

$4 \frac{\sqrt{x^{4}}}{\sqrt{29}}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$4 \frac{\sqrt{{(x^{2})}^{2}}}{\sqrt{29}}$

Step 7.2

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

$4 \frac{x^{2}}{\sqrt{29}}$

Step 8

Multiply $\frac{x^{2}}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$4 (\frac{x^{2}}{\sqrt{29}} \cdot \frac{\sqrt{29}}{\sqrt{29}})$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{x^{2}}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$4 \frac{x^{2} \sqrt{29}}{\sqrt{29} \sqrt{29}}$

Step 9.2

Raise $\sqrt{29}$ to the power of $1$ .

$4 \frac{x^{2} \sqrt{29}}{{\sqrt{29}}^{1} \sqrt{29}}$

Step 9.3

Raise $\sqrt{29}$ to the power of $1$ .

$4 \frac{x^{2} \sqrt{29}}{{\sqrt{29}}^{1} {\sqrt{29}}^{1}}$

Step 9.4

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

$4 \frac{x^{2} \sqrt{29}}{{\sqrt{29}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

$4 \frac{x^{2} \sqrt{29}}{{\sqrt{29}}^{2}}$

Step 9.6

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$4 \frac{x^{2} \sqrt{29}}{{(29^{\frac{1}{2}})}^{2}}$

Step 9.6.2

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

$4 \frac{x^{2} \sqrt{29}}{29^{\frac{1}{2} \cdot 2}}$

Step 9.6.3

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

$4 \frac{x^{2} \sqrt{29}}{29^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

$4 \frac{x^{2} \sqrt{29}}{29^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$4 \frac{x^{2} \sqrt{29}}{29^{1}}$

Step 9.6.5

Evaluate the exponent.

$4 \frac{x^{2} \sqrt{29}}{29}$

Step 10

Combine $4$ and $\frac{x^{2} \sqrt{29}}{29}$ .

$\frac{4 x^{2} \sqrt{29}}{29}$