Precalculus Examples

$\log ({(10)}^{\frac{\sqrt{x y^{3}}}{z^{4}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x y^{3}}$ as ${(x y^{3})}^{\frac{1}{2}}$ .

$\log (10^{\frac{{(x y^{3})}^{\frac{1}{2}}}{z^{4}}})$

Step 2

Expand $\log (10^{\frac{{(x y^{3})}^{\frac{1}{2}}}{z^{4}}})$ by moving $\frac{{(x y^{3})}^{\frac{1}{2}}}{z^{4}}$ outside the logarithm.

$\frac{{(x y^{3})}^{\frac{1}{2}}}{z^{4}} \log (10)$

Step 3

Logarithm base $10$ of $10$ is $1$ .

$\frac{{(x y^{3})}^{\frac{1}{2}}}{z^{4}} \cdot 1$

Step 4

Multiply $\frac{{(x y^{3})}^{\frac{1}{2}}}{z^{4}}$ by $1$ .

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the product rule to $x y^{3}$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

$\frac{x^{\frac{1}{2}} y^{\frac{3}{2}}}{z^{4}}$