Precalculus Examples

$\log_{c} (\sqrt[7]{\frac{x^{8}}{y^{7} z^{2}}})$

Step 1

Rewrite $\frac{x^{8}}{y^{7} z^{2}}$ as ${(\frac{x}{y})}^{7} \frac{x}{z^{2}}$ .

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Step 1.1

Factor the perfect power $x^{7}$ out of $x^{8}$ .

$\log_{c} (\sqrt[7]{\frac{x^{7} x}{y^{7} z^{2}}})$

Step 1.2

Rearrange the fraction $\frac{x^{7} x}{y^{7} z^{2}}$ .

$\log_{c} (\sqrt[7]{{(\frac{x}{y})}^{7} \frac{x}{z^{2}}})$

Step 2

Pull terms out from under the radical.

$\log_{c} (\frac{x}{y} \sqrt[7]{\frac{x}{z^{2}}})$

Step 3

Rewrite $\sqrt[7]{\frac{x}{z^{2}}}$ as $\frac{\sqrt[7]{x}}{\sqrt[7]{z^{2}}}$ .

$\log_{c} (\frac{x}{y} \cdot \frac{\sqrt[7]{x}}{\sqrt[7]{z^{2}}})$

Step 4

Combine.

$\log_{c} (\frac{x \sqrt[7]{x}}{y \sqrt[7]{z^{2}}})$

Step 5

Multiply $\frac{x \sqrt[7]{x}}{y \sqrt[7]{z^{2}}}$ by $\frac{{\sqrt[7]{z^{2}}}^{6}}{{\sqrt[7]{z^{2}}}^{6}}$ .

$\log_{c} (\frac{x \sqrt[7]{x}}{y \sqrt[7]{z^{2}}} \cdot \frac{{\sqrt[7]{z^{2}}}^{6}}{{\sqrt[7]{z^{2}}}^{6}})$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{x \sqrt[7]{x}}{y \sqrt[7]{z^{2}}}$ by $\frac{{\sqrt[7]{z^{2}}}^{6}}{{\sqrt[7]{z^{2}}}^{6}}$ .

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y \sqrt[7]{z^{2}} {\sqrt[7]{z^{2}}}^{6}})$

Step 6.2

Move $\sqrt[7]{z^{2}}$ .

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y (\sqrt[7]{z^{2}} {\sqrt[7]{z^{2}}}^{6})})$

Step 6.3

Raise $\sqrt[7]{z^{2}}$ to the power of $1$ .

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y ({\sqrt[7]{z^{2}}}^{1} {\sqrt[7]{z^{2}}}^{6})})$

Step 6.4

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

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y {\sqrt[7]{z^{2}}}^{1 + 6}})$

Step 6.5

Add $1$ and $6$ .

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y {\sqrt[7]{z^{2}}}^{7}})$

Step 6.6

Rewrite ${\sqrt[7]{z^{2}}}^{7}$ as $z^{2}$ .

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Step 6.6.1

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

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y {(z^{\frac{2}{7}})}^{7}})$

Step 6.6.2

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

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y z^{\frac{2}{7} \cdot 7}})$

Step 6.6.3

Combine $\frac{2}{7}$ and $7$ .

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y z^{\frac{2 \cdot 7}{7}}})$

Step 6.6.4

Multiply $2$ by $7$ .

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y z^{\frac{14}{7}}})$

Step 6.6.5

Cancel the common factor of $14$ and $7$ .

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Step 6.6.5.1

Factor $7$ out of $14$ .

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y z^{\frac{7 \cdot 2}{7}}})$

Step 6.6.5.2

Cancel the common factors.

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Step 6.6.5.2.1

Factor $7$ out of $7$ .

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y z^{\frac{7 \cdot 2}{7 (1)}}})$

Step 6.6.5.2.2

Cancel the common factor.

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y z^{\frac{7 \cdot 2}{7 \cdot 1}}})$

Step 6.6.5.2.3

Rewrite the expression.

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y z^{\frac{2}{1}}})$

Step 6.6.5.2.4

Divide $2$ by $1$ .

$\log_{c} (\frac{x \sqrt[7]{x} {\sqrt[7]{z^{2}}}^{6}}{y z^{2}})$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite ${\sqrt[7]{z^{2}}}^{6}$ as $\sqrt[7]{{(z^{2})}^{6}}$ .

$\log_{c} (\frac{x \sqrt[7]{x} \sqrt[7]{{(z^{2})}^{6}}}{y z^{2}})$

Step 7.2

Multiply the exponents in ${(z^{2})}^{6}$ .

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Step 7.2.1

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

$\log_{c} (\frac{x \sqrt[7]{x} \sqrt[7]{z^{2 \cdot 6}}}{y z^{2}})$

Step 7.2.2

Multiply $2$ by $6$ .

$\log_{c} (\frac{x \sqrt[7]{x} \sqrt[7]{z^{12}}}{y z^{2}})$

Step 7.3

Factor out $z^{7}$ .

$\log_{c} (\frac{x \sqrt[7]{x} \sqrt[7]{z^{7} z^{5}}}{y z^{2}})$

Step 7.4

Pull terms out from under the radical.

$\log_{c} (\frac{x \sqrt[7]{x} z \sqrt[7]{z^{5}}}{y z^{2}})$

Step 7.5

Combine using the product rule for radicals.

$\log_{c} (\frac{x z \sqrt[7]{x z^{5}}}{y z^{2}})$

Step 8

Cancel the common factor of $z$ and $z^{2}$ .

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Step 8.1

Factor $z$ out of $x z \sqrt[7]{x z^{5}}$ .

$\log_{c} (\frac{z (x \sqrt[7]{x z^{5}})}{y z^{2}})$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $z$ out of $y z^{2}$ .

$\log_{c} (\frac{z (x \sqrt[7]{x z^{5}})}{z (y z)})$

Step 8.2.2

Cancel the common factor.

$\log_{c} (\frac{z (x \sqrt[7]{x z^{5}})}{z (y z)})$

Step 8.2.3

Rewrite the expression.

$\log_{c} (\frac{x \sqrt[7]{x z^{5}}}{y z})$