Precalculus Examples

$\log ({(\frac{100 x^{2}}{\sqrt{y}})}^{\frac{3}{5}})$

Step 1

Multiply $\frac{100 x^{2}}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$\log ({(\frac{100 x^{2}}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}})}^{\frac{3}{5}})$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{100 x^{2}}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$\log ({(\frac{100 x^{2} \sqrt{y}}{\sqrt{y} \sqrt{y}})}^{\frac{3}{5}})$

Step 2.2

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

$\log ({(\frac{100 x^{2} \sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}})}^{\frac{3}{5}})$

Step 2.3

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

$\log ({(\frac{100 x^{2} \sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}})}^{\frac{3}{5}})$

Step 2.4

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

$\log ({(\frac{100 x^{2} \sqrt{y}}{{\sqrt{y}}^{1 + 1}})}^{\frac{3}{5}})$

Step 2.5

Add $1$ and $1$ .

$\log ({(\frac{100 x^{2} \sqrt{y}}{{\sqrt{y}}^{2}})}^{\frac{3}{5}})$

Step 2.6

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

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Step 2.6.1

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

$\log ({(\frac{100 x^{2} \sqrt{y}}{{(y^{\frac{1}{2}})}^{2}})}^{\frac{3}{5}})$

Step 2.6.2

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

$\log ({(\frac{100 x^{2} \sqrt{y}}{y^{\frac{1}{2} \cdot 2}})}^{\frac{3}{5}})$

Step 2.6.3

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

$\log ({(\frac{100 x^{2} \sqrt{y}}{y^{\frac{2}{2}}})}^{\frac{3}{5}})$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\log ({(\frac{100 x^{2} \sqrt{y}}{y^{\frac{2}{2}}})}^{\frac{3}{5}})$

Step 2.6.4.2

Rewrite the expression.

$\log ({(\frac{100 x^{2} \sqrt{y}}{y^{1}})}^{\frac{3}{5}})$

Step 2.6.5

Simplify.

$\log ({(\frac{100 x^{2} \sqrt{y}}{y})}^{\frac{3}{5}})$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $\frac{100 x^{2} \sqrt{y}}{y}$ .

$\log (\frac{{(100 x^{2} \sqrt{y})}^{\frac{3}{5}}}{y^{\frac{3}{5}}})$

Step 3.2

Apply the product rule to $100 x^{2} \sqrt{y}$ .

$\log (\frac{{(100 x^{2})}^{\frac{3}{5}} {\sqrt{y}}^{\frac{3}{5}}}{y^{\frac{3}{5}}})$

Step 3.3

Apply the product rule to $100 x^{2}$ .

$\log (\frac{100^{\frac{3}{5}} {(x^{2})}^{\frac{3}{5}} {\sqrt{y}}^{\frac{3}{5}}}{y^{\frac{3}{5}}})$

Step 4

Multiply the exponents in ${(x^{2})}^{\frac{3}{5}}$ .

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Step 4.1

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

$\log (\frac{100^{\frac{3}{5}} x^{2 (\frac{3}{5})} {\sqrt{y}}^{\frac{3}{5}}}{y^{\frac{3}{5}}})$

Step 4.2

Multiply $2 (\frac{3}{5})$ .

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Step 4.2.1

Combine $2$ and $\frac{3}{5}$ .

$\log (\frac{100^{\frac{3}{5}} x^{\frac{2 \cdot 3}{5}} {\sqrt{y}}^{\frac{3}{5}}}{y^{\frac{3}{5}}})$

Step 4.2.2

Multiply $2$ by $3$ .

$\log (\frac{100^{\frac{3}{5}} x^{\frac{6}{5}} {\sqrt{y}}^{\frac{3}{5}}}{y^{\frac{3}{5}}})$