Precalculus Examples

$\frac{120 (\ln ({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}}))}{{(16 z^{5})}^{\frac{1}{4}}}$

Step 1

Simplify $120 \ln ({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}})$ by moving $120$ inside the logarithm.

$\frac{\ln ({({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}})}^{120})}{{(16 z^{5})}^{\frac{1}{4}}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $16 z^{5}$ .

$\frac{\ln ({({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}})}^{120})}{16^{\frac{1}{4}} {(z^{5})}^{\frac{1}{4}}}$

Step 2.2

Rewrite $16$ as $2^{4}$ .

$\frac{\ln ({({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}})}^{120})}{{(2^{4})}^{\frac{1}{4}} {(z^{5})}^{\frac{1}{4}}}$

Step 2.3

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

$\frac{\ln ({({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}})}^{120})}{2^{4 (\frac{1}{4})} {(z^{5})}^{\frac{1}{4}}}$

Step 2.4

Cancel the common factor of $4$ .

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Step 2.4.1

Cancel the common factor.

$\frac{\ln ({({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}})}^{120})}{2^{4 (\frac{1}{4})} {(z^{5})}^{\frac{1}{4}}}$

Step 2.4.2

Rewrite the expression.

$\frac{\ln ({({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}})}^{120})}{2^{1} {(z^{5})}^{\frac{1}{4}}}$

Step 2.5

Evaluate the exponent.

$\frac{\ln ({({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}})}^{120})}{2 {(z^{5})}^{\frac{1}{4}}}$

Step 2.6

Multiply the exponents in ${(z^{5})}^{\frac{1}{4}}$ .

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Step 2.6.1

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

$\frac{\ln ({({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}})}^{120})}{2 z^{5 (\frac{1}{4})}}$

Step 2.6.2

Combine $5$ and $\frac{1}{4}$ .

$\frac{\ln ({({(x^{3})}^{\frac{1}{5}} {(y^{2})}^{\frac{1}{3}})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3

Simplify the numerator.

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Step 3.1

Multiply the exponents in ${(x^{3})}^{\frac{1}{5}}$ .

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Step 3.1.1

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

$\frac{\ln ({(x^{3 (\frac{1}{5})} {(y^{2})}^{\frac{1}{3}})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3.1.2

Combine $3$ and $\frac{1}{5}$ .

$\frac{\ln ({(x^{\frac{3}{5}} {(y^{2})}^{\frac{1}{3}})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3.2

Multiply the exponents in ${(y^{2})}^{\frac{1}{3}}$ .

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Step 3.2.1

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

$\frac{\ln ({(x^{\frac{3}{5}} y^{2 (\frac{1}{3})})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3.2.2

Combine $2$ and $\frac{1}{3}$ .

$\frac{\ln ({(x^{\frac{3}{5}} y^{\frac{2}{3}})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3.3

Apply the product rule to $x^{\frac{3}{5}} y^{\frac{2}{3}}$ .

$\frac{\ln ({(x^{\frac{3}{5}})}^{120} {(y^{\frac{2}{3}})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3.4

Multiply the exponents in ${(x^{\frac{3}{5}})}^{120}$ .

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Step 3.4.1

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

$\frac{\ln (x^{\frac{3}{5} \cdot 120} {(y^{\frac{2}{3}})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3.4.2

Cancel the common factor of $5$ .

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Step 3.4.2.1

Factor $5$ out of $120$ .

$\frac{\ln (x^{\frac{3}{5} \cdot (5 (24))} {(y^{\frac{2}{3}})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3.4.2.2

Cancel the common factor.

$\frac{\ln (x^{\frac{3}{5} \cdot (5 \cdot 24)} {(y^{\frac{2}{3}})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3.4.2.3

Rewrite the expression.

$\frac{\ln (x^{3 \cdot 24} {(y^{\frac{2}{3}})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3.4.3

Multiply $3$ by $24$ .

$\frac{\ln (x^{72} {(y^{\frac{2}{3}})}^{120})}{2 z^{\frac{5}{4}}}$

Step 3.5

Multiply the exponents in ${(y^{\frac{2}{3}})}^{120}$ .

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Step 3.5.1

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

$\frac{\ln (x^{72} y^{\frac{2}{3} \cdot 120})}{2 z^{\frac{5}{4}}}$

Step 3.5.2

Cancel the common factor of $3$ .

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Step 3.5.2.1

Factor $3$ out of $120$ .

$\frac{\ln (x^{72} y^{\frac{2}{3} \cdot (3 (40))})}{2 z^{\frac{5}{4}}}$

Step 3.5.2.2

Cancel the common factor.

$\frac{\ln (x^{72} y^{\frac{2}{3} \cdot (3 \cdot 40)})}{2 z^{\frac{5}{4}}}$

Step 3.5.2.3

Rewrite the expression.

$\frac{\ln (x^{72} y^{2 \cdot 40})}{2 z^{\frac{5}{4}}}$

Step 3.5.3

Multiply $2$ by $40$ .

$\frac{\ln (x^{72} y^{80})}{2 z^{\frac{5}{4}}}$