Basic Math Examples

$\frac{1}{\sqrt{f}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1

Simplify $\frac{1}{\sqrt{f}}$ .

Tap for more steps...

Step 1.1

Multiply $\frac{1}{\sqrt{f}}$ by $\frac{\sqrt{f}}{\sqrt{f}}$ .

$\frac{1}{\sqrt{f}} \cdot \frac{\sqrt{f}}{\sqrt{f}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2

Combine and simplify the denominator.

Tap for more steps...

Step 1.2.1

Multiply $\frac{1}{\sqrt{f}}$ by $\frac{\sqrt{f}}{\sqrt{f}}$ .

$\frac{\sqrt{f}}{\sqrt{f} \sqrt{f}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2.2

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

$\frac{\sqrt{f}}{{\sqrt{f}}^{1} \sqrt{f}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2.3

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

$\frac{\sqrt{f}}{{\sqrt{f}}^{1} {\sqrt{f}}^{1}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2.4

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

$\frac{\sqrt{f}}{{\sqrt{f}}^{1 + 1}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2.5

Add $1$ and $1$ .

$\frac{\sqrt{f}}{{\sqrt{f}}^{2}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2.6

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

Tap for more steps...

Step 1.2.6.1

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

$\frac{\sqrt{f}}{{(f^{\frac{1}{2}})}^{2}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2.6.2

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

$\frac{\sqrt{f}}{f^{\frac{1}{2} \cdot 2}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2.6.3

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

$\frac{\sqrt{f}}{f^{\frac{2}{2}}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.6.4.1

Cancel the common factor.

$\frac{\sqrt{f}}{f^{\frac{2}{2}}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2.6.4.2

Rewrite the expression.

$\frac{\sqrt{f}}{f^{1}} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 1.2.6.5

Simplify.

$\frac{\sqrt{f}}{f} = - 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 2

Simplify $- 2 \log (0.03 \cdot \frac{10^{- 3}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$ .

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Simplify the numerator.

Tap for more steps...

Step 2.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\sqrt{f}}{f} = - 2 \log (0.03 \cdot \frac{\frac{1}{10^{3}}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 2.1.1.2

Raise $10$ to the power of $3$ .

$\frac{\sqrt{f}}{f} = - 2 \log (0.03 \cdot \frac{\frac{1}{1000}}{3.7 \cdot 0.6} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 2.1.2

Multiply $3.7$ by $0.6$ .

$\frac{\sqrt{f}}{f} = - 2 \log (0.03 \cdot \frac{\frac{1}{1000}}{2.22} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 2.1.3

Cancel the common factor of $0.03$ .

Tap for more steps...

Step 2.1.3.1

Factor $0.03$ out of $2.22$ .

$\frac{\sqrt{f}}{f} = - 2 \log (0.03 \cdot \frac{\frac{1}{1000}}{0.03 (74)} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 2.1.3.2

Cancel the common factor.

$\frac{\sqrt{f}}{f} = - 2 \log (0.03 \cdot \frac{\frac{1}{1000}}{0.03 \cdot 74} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 2.1.3.3

Rewrite the expression.

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{\frac{1}{1000}}{74} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 2.1.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{1000} \cdot \frac{1}{74} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 2.1.5

Multiply $\frac{1}{1000} \cdot \frac{1}{74}$ .

Tap for more steps...

Step 2.1.5.1

Multiply $\frac{1}{1000}$ by $\frac{1}{74}$ .

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{1000 \cdot 74} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 2.1.5.2

Multiply $1000$ by $74$ .

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51}{902700 \cdot \sqrt{f}})$

Step 2.1.6

Multiply $\frac{2.51}{902700 \cdot \sqrt{f}}$ by $\frac{\sqrt{f}}{\sqrt{f}}$ .

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51}{902700 \cdot \sqrt{f}} \cdot \frac{\sqrt{f}}{\sqrt{f}})$

Step 2.1.7

Combine and simplify the denominator.

Tap for more steps...

Step 2.1.7.1

Multiply $\frac{2.51}{902700 \cdot \sqrt{f}}$ by $\frac{\sqrt{f}}{\sqrt{f}}$ .

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 \cdot \sqrt{f} \sqrt{f}})$

Step 2.1.7.2

Move $\sqrt{f}$ .

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 (\sqrt{f} \sqrt{f})})$

Step 2.1.7.3

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

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 ({\sqrt{f}}^{1} \sqrt{f})})$

Step 2.1.7.4

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

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 ({\sqrt{f}}^{1} {\sqrt{f}}^{1})})$

Step 2.1.7.5

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

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 {\sqrt{f}}^{1 + 1}})$

Step 2.1.7.6

Add $1$ and $1$ .

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 {\sqrt{f}}^{2}})$

Step 2.1.7.7

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

Tap for more steps...

Step 2.1.7.7.1

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

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 {(f^{\frac{1}{2}})}^{2}})$

Step 2.1.7.7.2

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

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 f^{\frac{1}{2} \cdot 2}})$

Step 2.1.7.7.3

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

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 f^{\frac{2}{2}}})$

Step 2.1.7.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.7.7.4.1

Cancel the common factor.

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 f^{\frac{2}{2}}})$

Step 2.1.7.7.4.2

Rewrite the expression.

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 f^{1}})$

Step 2.1.7.7.5

Simplify.

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 \sqrt{f}}{902700 f})$

Step 2.1.8

Factor $2.51$ out of $2.51 \sqrt{f}$ .

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 (\sqrt{f})}{902700 f})$

Step 2.1.9

Factor $902700$ out of $902700 f$ .

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51 (\sqrt{f})}{902700 (f)})$

Step 2.1.10

Separate fractions.

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{2.51}{902700} \cdot \frac{\sqrt{f}}{f})$

Step 2.1.11

Divide $2.51$ by $902700$ .

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + 0.00000278 \frac{\sqrt{f}}{f})$

Step 2.1.12

Combine $0.00000278$ and $\frac{\sqrt{f}}{f}$ .

$\frac{\sqrt{f}}{f} = - 2 \log (\frac{1}{74000} + \frac{0.00000278 \sqrt{f}}{f})$

Step 2.2

Simplify $- 2 \log (\frac{1}{74000} + \frac{0.00000278 \sqrt{f}}{f})$ by moving $2$ inside the logarithm.

$\frac{\sqrt{f}}{f} = - \log ({(\frac{1}{74000} + \frac{0.00000278 \sqrt{f}}{f})}^{2})$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$f \approx 0.01280126$

Step 4