Finite Math Examples

$\frac{4}{3} \cdot (p {(\frac{1}{2} t + 2)}^{3}) = 36 {pm}^{3}$

Step 1

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} (\frac{4}{3} \cdot (p {(\frac{1}{2} t + 2)}^{3})) = \frac{3}{4} (36 {pm}^{3})$

Step 2

Simplify both sides of the equation.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Simplify $\frac{3}{4} (\frac{4}{3} \cdot (p {(\frac{1}{2} t + 2)}^{3}))$ .

Tap for more steps...

Step 2.1.1.1

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

$\frac{3}{4} (\frac{4}{3} \cdot (p {(\frac{t}{2} + 2)}^{3})) = \frac{3}{4} (36 {pm}^{3})$

Step 2.1.1.2

Multiply $\frac{4}{3} (p {(\frac{t}{2} + 2)}^{3})$ .

Tap for more steps...

Step 2.1.1.2.1

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

$\frac{3}{4} (\frac{p \cdot 4}{3} {(\frac{t}{2} + 2)}^{3}) = \frac{3}{4} (36 {pm}^{3})$

Step 2.1.1.2.2

Combine $\frac{p \cdot 4}{3}$ and ${(\frac{t}{2} + 2)}^{3}$ .

$\frac{3}{4} \cdot \frac{p \cdot 4 {(\frac{t}{2} + 2)}^{3}}{3} = \frac{3}{4} (36 {pm}^{3})$

Step 2.1.1.3

Combine.

$\frac{3 (p \cdot 4 {(\frac{t}{2} + 2)}^{3})}{4 \cdot 3} = \frac{3}{4} (36 {pm}^{3})$

Step 2.1.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.1.1.4.1

Cancel the common factor.

$\frac{3 (p \cdot 4 {(\frac{t}{2} + 2)}^{3})}{4 \cdot 3} = \frac{3}{4} (36 {pm}^{3})$

Step 2.1.1.4.2

Rewrite the expression.

$\frac{p \cdot 4 {(\frac{t}{2} + 2)}^{3}}{4} = \frac{3}{4} (36 {pm}^{3})$

Step 2.1.1.5

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.1.1.5.1

Cancel the common factor.

$\frac{p \cdot 4 {(\frac{t}{2} + 2)}^{3}}{4} = \frac{3}{4} (36 {pm}^{3})$

Step 2.1.1.5.2

Divide $p {(\frac{t}{2} + 2)}^{3}$ by $1$ .

$p {(\frac{t}{2} + 2)}^{3} = \frac{3}{4} (36 {pm}^{3})$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Combine $\frac{3}{4}$ and $36 {pm}^{3}$ .

$p {(\frac{t}{2} + 2)}^{3} = \frac{3 (36 {pm}^{3})}{4}$

Step 3

Divide each term in $p {(\frac{t}{2} + 2)}^{3} = \frac{3 (36 {pm}^{3})}{4}$ by $p$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $p {(\frac{t}{2} + 2)}^{3} = \frac{3 (36 {pm}^{3})}{4}$ by $p$ .

$\frac{p {(\frac{t}{2} + 2)}^{3}}{p} = \frac{\frac{3 (36 {pm}^{3})}{4}}{p}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $p$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{p {(\frac{t}{2} + 2)}^{3}}{p} = \frac{\frac{3 (36 {pm}^{3})}{4}}{p}$

Step 3.2.1.2

Divide ${(\frac{t}{2} + 2)}^{3}$ by $1$ .

${(\frac{t}{2} + 2)}^{3} = \frac{\frac{3 (36 {pm}^{3})}{4}}{p}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

${(\frac{t}{2} + 2)}^{3} = \frac{3 (36 {pm}^{3})}{4} \cdot \frac{1}{p}$

Step 3.3.2

Multiply $\frac{3 (36 {pm}^{3})}{4}$ by $\frac{1}{p}$ .

${(\frac{t}{2} + 2)}^{3} = \frac{3 (36 {pm}^{3})}{4 p}$

Step 4

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

$\frac{t}{2} + 2 = \sqrt[3]{\frac{3 (36 {pm}^{3})}{4 p}}$

Step 5

Simplify $\sqrt[3]{\frac{3 (36 {pm}^{3})}{4 p}}$ .

Tap for more steps...

Step 5.1

Rewrite $\sqrt[3]{\frac{3 (36 {pm}^{3})}{4 p}}$ as $\frac{\sqrt[3]{3 (36 {pm}^{3})}}{\sqrt[3]{4 p}}$ .

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})}}{\sqrt[3]{4 p}}$

Step 5.2

Multiply $\frac{\sqrt[3]{3 (36 {pm}^{3})}}{\sqrt[3]{4 p}}$ by $\frac{{\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{2}}$ .

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})}}{\sqrt[3]{4 p}} \cdot \frac{{\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{2}}$

Step 5.3

Combine and simplify the denominator.

Tap for more steps...

Step 5.3.1

Multiply $\frac{\sqrt[3]{3 (36 {pm}^{3})}}{\sqrt[3]{4 p}}$ by $\frac{{\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{2}}$ .

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} {\sqrt[3]{4 p}}^{2}}{\sqrt[3]{4 p} {\sqrt[3]{4 p}}^{2}}$

Step 5.3.2

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

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} {\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{1} {\sqrt[3]{4 p}}^{2}}$

Step 5.3.3

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

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} {\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{1 + 2}}$

Step 5.3.4

Add $1$ and $2$ .

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} {\sqrt[3]{4 p}}^{2}}{{\sqrt[3]{4 p}}^{3}}$

Step 5.3.5

Rewrite ${\sqrt[3]{4 p}}^{3}$ as $4 p$ .

Tap for more steps...

Step 5.3.5.1

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

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} {\sqrt[3]{4 p}}^{2}}{{({(4 p)}^{\frac{1}{3}})}^{3}}$

Step 5.3.5.2

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

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{\frac{1}{3} \cdot 3}}$

Step 5.3.5.3

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

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{\frac{3}{3}}}$

Step 5.3.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.3.5.4.1

Cancel the common factor.

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{\frac{3}{3}}}$

Step 5.3.5.4.2

Rewrite the expression.

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} {\sqrt[3]{4 p}}^{2}}{{(4 p)}^{1}}$

Step 5.3.5.5

Simplify.

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} {\sqrt[3]{4 p}}^{2}}{4 p}$

Step 5.4

Simplify the numerator.

Tap for more steps...

Step 5.4.1

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

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} \sqrt[3]{{(4 p)}^{2}}}{4 p}$

Step 5.4.2

Apply the product rule to $4 p$ .

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} \sqrt[3]{4^{2} p^{2}}}{4 p}$

Step 5.4.3

Raise $4$ to the power of $2$ .

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} \sqrt[3]{16 p^{2}}}{4 p}$

Step 5.4.4

Rewrite $16 p^{2}$ as $2^{3} \cdot (2 p^{2})$ .

Tap for more steps...

Step 5.4.4.1

Factor $8$ out of $16$ .

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} \sqrt[3]{8 (2) p^{2}}}{4 p}$

Step 5.4.4.2

Rewrite $8$ as $2^{3}$ .

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} \sqrt[3]{2^{3} \cdot 2 p^{2}}}{4 p}$

Step 5.4.4.3

Add parentheses.

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} \sqrt[3]{2^{3} \cdot (2 p^{2})}}{4 p}$

Step 5.4.5

Pull terms out from under the radical.

$\frac{t}{2} + 2 = \frac{\sqrt[3]{3 (36 {pm}^{3})} \cdot 2 \sqrt[3]{2 p^{2}}}{4 p}$

Step 5.4.6

Combine exponents.

Tap for more steps...

Step 5.4.6.1

Combine using the product rule for radicals.

$\frac{t}{2} + 2 = \frac{2 \sqrt[3]{3 (36 {pm}^{3}) (2 p^{2})}}{4 p}$

Step 5.4.6.2

Multiply $2$ by $3$ .

$\frac{t}{2} + 2 = \frac{2 \sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{4 p}$

Step 5.5

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 5.5.1

Factor $2$ out of $2 \sqrt[3]{6 (36 {pm}^{3}) p^{2}}$ .

$\frac{t}{2} + 2 = \frac{2 \sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{4 p}$

Step 5.5.2

Cancel the common factors.

Tap for more steps...

Step 5.5.2.1

Factor $2$ out of $4 p$ .

$\frac{t}{2} + 2 = \frac{2 \sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 (2 p)}$

Step 5.5.2.2

Cancel the common factor.

$\frac{t}{2} + 2 = \frac{2 \sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 (2 p)}$

Step 5.5.2.3

Rewrite the expression.

$\frac{t}{2} + 2 = \frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 p}$

Step 6

Subtract $2$ from both sides of the equation.

$\frac{t}{2} = \frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 p} - 2$

Step 7

Multiply both sides of the equation by $2$ .

$2 \frac{t}{2} = 2 (\frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 p} - 2)$

Step 8

Simplify both sides of the equation.

Tap for more steps...

Step 8.1

Simplify the left side.

Tap for more steps...

Step 8.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.1.1.1

Cancel the common factor.

$2 \frac{t}{2} = 2 (\frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 p} - 2)$

Step 8.1.1.2

Rewrite the expression.

$t = 2 (\frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 p} - 2)$

Step 8.2

Simplify the right side.

Tap for more steps...

Step 8.2.1

Simplify $2 (\frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 p} - 2)$ .

Tap for more steps...

Step 8.2.1.1

Apply the distributive property.

$t = 2 \frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 p} + 2 \cdot - 2$

Step 8.2.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.2.1.2.1

Factor $2$ out of $2 p$ .

$t = 2 \frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 (p)} + 2 \cdot - 2$

Step 8.2.1.2.2

Cancel the common factor.

$t = 2 \frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{2 p} + 2 \cdot - 2$

Step 8.2.1.2.3

Rewrite the expression.

$t = \frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{p} + 2 \cdot - 2$

Step 8.2.1.3

Multiply $2$ by $- 2$ .

$t = \frac{\sqrt[3]{6 (36 {pm}^{3}) p^{2}}}{p} - 4$

Step 9

Move $36 {pm}^{3}$ .

$t = \frac{\sqrt[3]{6 p^{2} (36 {pm}^{3})}}{p} - 4$

Step 10

Simplify each term.

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Multiply $36$ by $6$ .

$t = \frac{\sqrt[3]{216 p^{2} {pm}^{3}}}{p} - 4$

Step 10.1.2

Rewrite $216 p^{2} {pm}^{3}$ as ${(6 pm)}^{3} p^{2}$ .

Tap for more steps...

Step 10.1.2.1

Rewrite $216$ as $6^{3}$ .

$t = \frac{\sqrt[3]{6^{3} p^{2} {pm}^{3}}}{p} - 4$

Step 10.1.2.2

Move $p^{2}$ .

$t = \frac{\sqrt[3]{6^{3} {pm}^{3} p^{2}}}{p} - 4$

Step 10.1.2.3

Rewrite $6^{3} {pm}^{3}$ as ${(6 pm)}^{3}$ .

$t = \frac{\sqrt[3]{{(6 pm)}^{3} p^{2}}}{p} - 4$

Step 10.1.3

Pull terms out from under the radical.

$t = \frac{6 pm \sqrt[3]{p^{2}}}{p} - 4$

Step 10.2

Rewrite the expression.

$t = \frac{6 \sqrt[3]{p^{2}}}{p} pm - 4$