Precalculus Examples

$\log_{b} (x^{2}) \cdot \sqrt{\frac{y}{t}}$

Step 1

Expand $\log_{b} (x^{2})$ by moving $2$ outside the logarithm.

$2 \log_{b} (x) \cdot \sqrt{\frac{y}{t}}$

Step 2

Rewrite $\sqrt{\frac{y}{t}}$ as $\frac{\sqrt{y}}{\sqrt{t}}$ .

$2 \log_{b} (x) \cdot \frac{\sqrt{y}}{\sqrt{t}}$

Step 3

Multiply $\frac{\sqrt{y}}{\sqrt{t}}$ by $\frac{\sqrt{t}}{\sqrt{t}}$ .

$2 \log_{b} (x) \cdot (\frac{\sqrt{y}}{\sqrt{t}} \cdot \frac{\sqrt{t}}{\sqrt{t}})$

Step 4

Combine and simplify the denominator.

Tap for more steps...

Step 4.1

Multiply $\frac{\sqrt{y}}{\sqrt{t}}$ by $\frac{\sqrt{t}}{\sqrt{t}}$ .

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{\sqrt{t} \sqrt{t}}$

Step 4.2

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

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{{\sqrt{t}}^{1} \sqrt{t}}$

Step 4.3

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

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{{\sqrt{t}}^{1} {\sqrt{t}}^{1}}$

Step 4.4

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

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{{\sqrt{t}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{{\sqrt{t}}^{2}}$

Step 4.6

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

Tap for more steps...

Step 4.6.1

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

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{{(t^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{t^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{t^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.6.4.1

Cancel the common factor.

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{t^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{t^{1}}$

Step 4.6.5

Simplify.

$2 \log_{b} (x) \cdot \frac{\sqrt{y} \sqrt{t}}{t}$

Step 5

Combine using the product rule for radicals.

$2 \log_{b} (x) \cdot \frac{\sqrt{y t}}{t}$

Step 6

Multiply $2 \log_{b} (x) \frac{\sqrt{y t}}{t}$ .

Tap for more steps...

Step 6.1

Combine $\frac{\sqrt{y t}}{t}$ and $2$ .

$\frac{\sqrt{y t} \cdot 2}{t} \log_{b} (x)$

Step 6.2

Combine $\frac{\sqrt{y t} \cdot 2}{t}$ and $\log_{b} (x)$ .

$\frac{\sqrt{y t} \cdot 2 \log_{b} (x)}{t}$

Step 7

Move $2$ to the left of $\sqrt{y t}$ .

$\frac{2 \sqrt{y t} \log_{b} (x)}{t}$