Basic Math Examples

$\frac{1}{4^{p^{3}}} - \frac{1}{3} b = 4$

Step 1

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

$\frac{1}{4^{p^{3}}} - \frac{b}{3} = 4$

Step 2

Add $\frac{b}{3}$ to both sides of the equation.

$\frac{1}{4^{p^{3}}} = 4 + \frac{b}{3}$

Step 3

Multiply both sides of the equation by $4^{p^{3}}$ .

$1 = 4^{p^{3}} (4 + \frac{b}{3})$

Step 4

Rewrite the equation as $4^{p^{3}} (4 + \frac{b}{3}) = 1$ .

$4^{p^{3}} (4 + \frac{b}{3}) = 1$

Step 5

Divide each term in $4^{p^{3}} (4 + \frac{b}{3}) = 1$ by $4 + \frac{b}{3}$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $4^{p^{3}} (4 + \frac{b}{3}) = 1$ by $4 + \frac{b}{3}$ .

$\frac{4^{p^{3}} (4 + \frac{b}{3})}{4 + \frac{b}{3}} = \frac{1}{4 + \frac{b}{3}}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $4 + \frac{b}{3}$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$\frac{4^{p^{3}} (4 + \frac{b}{3})}{4 + \frac{b}{3}} = \frac{1}{4 + \frac{b}{3}}$

Step 5.2.1.2

Divide $4^{p^{3}}$ by $1$ .

$4^{p^{3}} = \frac{1}{4 + \frac{b}{3}}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Simplify the denominator.

Tap for more steps...

Step 5.3.1.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4^{p^{3}} = \frac{1}{4 \cdot \frac{3}{3} + \frac{b}{3}}$

Step 5.3.1.2

Combine $4$ and $\frac{3}{3}$ .

$4^{p^{3}} = \frac{1}{\frac{4 \cdot 3}{3} + \frac{b}{3}}$

Step 5.3.1.3

Combine the numerators over the common denominator.

$4^{p^{3}} = \frac{1}{\frac{4 \cdot 3 + b}{3}}$

Step 5.3.1.4

Multiply $4$ by $3$ .

$4^{p^{3}} = \frac{1}{\frac{12 + b}{3}}$

Step 5.3.2

Multiply the numerator by the reciprocal of the denominator.

$4^{p^{3}} = 1 \frac{3}{12 + b}$

Step 5.3.3

Multiply $\frac{3}{12 + b}$ by $1$ .

$4^{p^{3}} = \frac{3}{12 + b}$

Step 6

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (4^{p^{3}}) = \ln (\frac{3}{12 + b})$

Step 7

Expand $\ln (4^{p^{3}})$ by moving $p^{3}$ outside the logarithm.

$p^{3} \ln (4) = \ln (\frac{3}{12 + b})$

Step 8

Divide each term in $p^{3} \ln (4) = \ln (\frac{3}{12 + b})$ by $\ln (4)$ and simplify.

Tap for more steps...

Step 8.1

Divide each term in $p^{3} \ln (4) = \ln (\frac{3}{12 + b})$ by $\ln (4)$ .

$\frac{p^{3} \ln (4)}{\ln (4)} = \frac{\ln (\frac{3}{12 + b})}{\ln (4)}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor of $\ln (4)$ .

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{p^{3} \ln (4)}{\ln (4)} = \frac{\ln (\frac{3}{12 + b})}{\ln (4)}$

Step 8.2.1.2

Divide $p^{3}$ by $1$ .

$p^{3} = \frac{\ln (\frac{3}{12 + b})}{\ln (4)}$

Step 9

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$p = \sqrt[3]{\frac{\ln (\frac{3}{12 + b})}{\ln (4)}}$

Step 10

Simplify $\sqrt[3]{\frac{\ln (\frac{3}{12 + b})}{\ln (4)}}$ .

Tap for more steps...

Step 10.1

Rewrite $\sqrt[3]{\frac{\ln (\frac{3}{12 + b})}{\ln (4)}}$ as $\frac{\sqrt[3]{\ln (\frac{3}{12 + b})}}{\sqrt[3]{\ln (4)}}$ .

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})}}{\sqrt[3]{\ln (4)}}$

Step 10.2

Multiply $\frac{\sqrt[3]{\ln (\frac{3}{12 + b})}}{\sqrt[3]{\ln (4)}}$ by $\frac{{\sqrt[3]{\ln (4)}}^{2}}{{\sqrt[3]{\ln (4)}}^{2}}$ .

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})}}{\sqrt[3]{\ln (4)}} \cdot \frac{{\sqrt[3]{\ln (4)}}^{2}}{{\sqrt[3]{\ln (4)}}^{2}}$

Step 10.3

Combine and simplify the denominator.

Tap for more steps...

Step 10.3.1

Multiply $\frac{\sqrt[3]{\ln (\frac{3}{12 + b})}}{\sqrt[3]{\ln (4)}}$ by $\frac{{\sqrt[3]{\ln (4)}}^{2}}{{\sqrt[3]{\ln (4)}}^{2}}$ .

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} {\sqrt[3]{\ln (4)}}^{2}}{\sqrt[3]{\ln (4)} {\sqrt[3]{\ln (4)}}^{2}}$

Step 10.3.2

Raise $\sqrt[3]{\ln (4)}$ to the power of $1$ .

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} {\sqrt[3]{\ln (4)}}^{2}}{{\sqrt[3]{\ln (4)}}^{1} {\sqrt[3]{\ln (4)}}^{2}}$

Step 10.3.3

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

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} {\sqrt[3]{\ln (4)}}^{2}}{{\sqrt[3]{\ln (4)}}^{1 + 2}}$

Step 10.3.4

Add $1$ and $2$ .

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} {\sqrt[3]{\ln (4)}}^{2}}{{\sqrt[3]{\ln (4)}}^{3}}$

Step 10.3.5

Rewrite ${\sqrt[3]{\ln (4)}}^{3}$ as $\ln (4)$ .

Tap for more steps...

Step 10.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\ln (4)}$ as ${\ln (4)}^{\frac{1}{3}}$ .

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} {\sqrt[3]{\ln (4)}}^{2}}{{({\ln (4)}^{\frac{1}{3}})}^{3}}$

Step 10.3.5.2

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

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} {\sqrt[3]{\ln (4)}}^{2}}{{\ln (4)}^{\frac{1}{3} \cdot 3}}$

Step 10.3.5.3

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

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} {\sqrt[3]{\ln (4)}}^{2}}{{\ln (4)}^{\frac{3}{3}}}$

Step 10.3.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 10.3.5.4.1

Cancel the common factor.

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} {\sqrt[3]{\ln (4)}}^{2}}{{\ln (4)}^{\frac{3}{3}}}$

Step 10.3.5.4.2

Rewrite the expression.

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} {\sqrt[3]{\ln (4)}}^{2}}{\ln^{1} (4)}$

Step 10.3.5.5

Simplify.

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} {\sqrt[3]{\ln (4)}}^{2}}{\ln (4)}$

Step 10.4

Rewrite ${\sqrt[3]{\ln (4)}}^{2}$ as $\sqrt[3]{\ln^{2} (4)}$ .

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b})} \sqrt[3]{\ln^{2} (4)}}{\ln (4)}$

Step 10.5

Combine using the product rule for radicals.

$p = \frac{\sqrt[3]{\ln (\frac{3}{12 + b}) \ln^{2} (4)}}{\ln (4)}$