Algebra Examples

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

Step 1

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

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Step 1.1

Subtract $29$ from both sides of the equation.

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

Step 1.2

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.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Cancel the common factor of $2$ .

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Step 3.1.2.2.1

Cancel the common factor.

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

Step 3.1.2.2.2

Rewrite the expression.

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

Step 3.1.2.3

Cancel the common factor of $3$ .

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Step 3.1.2.3.1

Cancel the common factor.

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

Step 3.1.2.3.2

Rewrite the expression.

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

Step 3.1.3

Simplify.

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

Step 3.1.4

Apply the distributive property.

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

Step 3.1.5

Simplify the expression.

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Step 3.1.5.1

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

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

Step 3.1.5.2

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.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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.

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Step 4.3.1

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

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor.

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

Step 4.3.2.2

Divide $y$ by $1$ .

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

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

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

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

Step 4.3.3.1.2

Divide $- 48$ by $- 3$ .

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

Step 4.3.3.1.3

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

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

Step 4.3.3.1.4

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

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

Step 4.3.3.1.5

Cancel the common factor of $2$ .

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Step 4.3.3.1.5.1

Cancel the common factor.

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

Step 4.3.3.1.5.2

Rewrite the expression.

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

Step 4.3.3.1.6

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

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

Step 4.3.3.1.7

Cancel the common factor.

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

Step 4.3.3.1.8

Divide $2$ by $1$ .

$y = 64 + 2$

Step 4.3.3.2

Add $64$ and $2$ .

$y = 66$

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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.

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Step 4.6.1

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

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

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor.

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

Step 4.6.2.2

Divide $y$ by $1$ .

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

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Simplify each term.

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Step 4.6.3.1.1

Move the negative in front of the fraction.

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

Step 4.6.3.1.2

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

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

Step 4.6.3.1.3

Divide $- 48$ by $- 3$ .

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

Step 4.6.3.1.4

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

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

Step 4.6.3.1.5

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

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

Step 4.6.3.1.6

Cancel the common factor of $2$ .

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Step 4.6.3.1.6.1

Cancel the common factor.

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

Step 4.6.3.1.6.2

Rewrite the expression.

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

Step 4.6.3.1.7

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

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

Step 4.6.3.1.8

Multiply $- 1$ by $64$ .

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

Step 4.6.3.1.9

Cancel the common factor.

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

Step 4.6.3.1.10

Divide $2$ by $1$ .

$y = - 64 + 2$

Step 4.6.3.2

Add $- 64$ and $2$ .

$y = - 62$

Step 4.7

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

$y = 66, - 62$