Solve for y -3(y-2)^(2/3)+29=-19

$- 3 {(y - 2)}^{\frac{2}{3}} + 29 = - 19$

Step 1

Move all terms not containing $y$ to the right side of the equation.

Subtract $29$ from both sides of the equation.

$- 3 {(y - 2)}^{\frac{2}{3}} = - 19 - 29$

Subtract $29$ from $- 19$ .

$- 3 {(y - 2)}^{\frac{2}{3}} = - 48$

Step 2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(- 3 {(y - 2)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(- 48)}^{\frac{3}{2}}$

Step 3

Simplify the left side.

Simplify ${(- 3 {(y - 2)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Apply the product rule to $- 3 {(y - 2)}^{\frac{2}{3}}$ .

${(- 3)}^{\frac{3}{2}} {({(y - 2)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(- 48)}^{\frac{3}{2}}$

Multiply the exponents in ${({(y - 2)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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

${(- 3)}^{\frac{3}{2}} {(y - 2)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(- 48)}^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- 3)}^{\frac{3}{2}} {(y - 2)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(- 48)}^{\frac{3}{2}}$

Rewrite the expression.

${(- 3)}^{\frac{3}{2}} {(y - 2)}^{\frac{1}{3} \cdot 3} = \pm {(- 48)}^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(- 3)}^{\frac{3}{2}} {(y - 2)}^{\frac{1}{3} \cdot 3} = \pm {(- 48)}^{\frac{3}{2}}$

Rewrite the expression.

${(- 3)}^{\frac{3}{2}} {(y - 2)}^{1} = \pm {(- 48)}^{\frac{3}{2}}$

Simplify.

${(- 3)}^{\frac{3}{2}} (y - 2) = \pm {(- 48)}^{\frac{3}{2}}$

Apply the distributive property.

${(- 3)}^{\frac{3}{2}} y + {(- 3)}^{\frac{3}{2}} \cdot - 2 = \pm {(- 48)}^{\frac{3}{2}}$

Simplify the expression.

Move $- 2$ to the left of ${(- 3)}^{\frac{3}{2}}$ .

${(- 3)}^{\frac{3}{2}} y - 2 \cdot {(- 3)}^{\frac{3}{2}} = \pm {(- 48)}^{\frac{3}{2}}$

Reorder factors in ${(- 3)}^{\frac{3}{2}} y - 2 {(- 3)}^{\frac{3}{2}}$ .

$y {(- 3)}^{\frac{3}{2}} - 2 {(- 3)}^{\frac{3}{2}} = \pm {(- 48)}^{\frac{3}{2}}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y {(- 3)}^{\frac{3}{2}} - 2 {(- 3)}^{\frac{3}{2}} = {(- 48)}^{\frac{3}{2}}$

Add $2 {(- 3)}^{\frac{3}{2}}$ to both sides of the equation.

$y {(- 3)}^{\frac{3}{2}} = {(- 48)}^{\frac{3}{2}} + 2 {(- 3)}^{\frac{3}{2}}$

Divide each term in $y {(- 3)}^{\frac{3}{2}} = {(- 48)}^{\frac{3}{2}} + 2 {(- 3)}^{\frac{3}{2}}$ by ${(- 3)}^{\frac{3}{2}}$ and simplify.

Divide each term in $y {(- 3)}^{\frac{3}{2}} = {(- 48)}^{\frac{3}{2}} + 2 {(- 3)}^{\frac{3}{2}}$ by ${(- 3)}^{\frac{3}{2}}$ .

$\frac{y {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} = \frac{{(- 48)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Simplify the left side.

Cancel the common factor.

$\frac{y {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} = \frac{{(- 48)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Divide $y$ by $1$ .

$y = \frac{{(- 48)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Simplify the right side.

Simplify each term.

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$y = {(\frac{- 48}{- 3})}^{\frac{3}{2}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Divide $- 48$ by $- 3$ .

$y = 16^{\frac{3}{2}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Rewrite $16$ as $4^{2}$ .

$y = {(4^{2})}^{\frac{3}{2}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

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

$y = 4^{2 (\frac{3}{2})} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = 4^{2 (\frac{3}{2})} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Rewrite the expression.

$y = 4^{3} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Raise $4$ to the power of $3$ .

$y = 64 + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Cancel the common factor.

$y = 64 + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Divide $2$ by $1$ .

$y = 64 + 2$

Add $64$ and $2$ .

$y = 66$

Next, use the negative value of the $\pm$ to find the second solution.

$y {(- 3)}^{\frac{3}{2}} - 2 {(- 3)}^{\frac{3}{2}} = - {(- 48)}^{\frac{3}{2}}$

Add $2 {(- 3)}^{\frac{3}{2}}$ to both sides of the equation.

$y {(- 3)}^{\frac{3}{2}} = - {(- 48)}^{\frac{3}{2}} + 2 {(- 3)}^{\frac{3}{2}}$

Divide each term in $y {(- 3)}^{\frac{3}{2}} = - {(- 48)}^{\frac{3}{2}} + 2 {(- 3)}^{\frac{3}{2}}$ by ${(- 3)}^{\frac{3}{2}}$ and simplify.

Divide each term in $y {(- 3)}^{\frac{3}{2}} = - {(- 48)}^{\frac{3}{2}} + 2 {(- 3)}^{\frac{3}{2}}$ by ${(- 3)}^{\frac{3}{2}}$ .

$\frac{y {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} = \frac{- {(- 48)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Simplify the left side.

Cancel the common factor.

$\frac{y {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} = \frac{- {(- 48)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Divide $y$ by $1$ .

$y = \frac{- {(- 48)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$y = - \frac{{(- 48)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$y = - {(\frac{- 48}{- 3})}^{\frac{3}{2}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Divide $- 48$ by $- 3$ .

$y = - 16^{\frac{3}{2}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Rewrite $16$ as $4^{2}$ .

$y = - {(4^{2})}^{\frac{3}{2}} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

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

$y = - 4^{2 (\frac{3}{2})} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$y = - 4^{2 (\frac{3}{2})} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Rewrite the expression.

$y = - 4^{3} + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Raise $4$ to the power of $3$ .

$y = - 1 \cdot 64 + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Multiply $- 1$ by $64$ .

$y = - 64 + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Cancel the common factor.

$y = - 64 + \frac{2 {(- 3)}^{\frac{3}{2}}}{{(- 3)}^{\frac{3}{2}}}$

Divide $2$ by $1$ .

$y = - 64 + 2$

Add $- 64$ and $2$ .

$y = - 62$

The complete solution is the result of both the positive and negative portions of the solution.

$y = 66, - 62$

Solve for y -3(y-2)^(2/3)+29=-19

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