Solve by Factoring 3(x+1)^(4/3)=48

$3 {(x + 1)}^{\frac{4}{3}} = 48$

Step 1

Subtract $48$ from both sides of the equation.

$3 {(x + 1)}^{\frac{4}{3}} - 48 = 0$

Step 2

Factor $3$ out of $3 {(x + 1)}^{\frac{4}{3}} - 48$ .

Factor $3$ out of $3 {(x + 1)}^{\frac{4}{3}}$ .

$3 ({(x + 1)}^{\frac{4}{3}}) - 48 = 0$

Factor $3$ out of $- 48$ .

$3 {(x + 1)}^{\frac{4}{3}} + 3 \cdot - 16 = 0$

Factor $3$ out of $3 {(x + 1)}^{\frac{4}{3}} + 3 \cdot - 16$ .

$3 ({(x + 1)}^{\frac{4}{3}} - 16) = 0$

Step 3

Rewrite ${(x + 1)}^{\frac{4}{3}}$ as ${({(x + 1)}^{\frac{2}{3}})}^{2}$ .

$3 ({({(x + 1)}^{\frac{2}{3}})}^{2} - 16) = 0$

Step 4

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

$3 ({({(x + 1)}^{\frac{2}{3}})}^{2} - 4^{2}) = 0$

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x + 1)}^{\frac{2}{3}}$ and $b = 4$ .

$3 (({(x + 1)}^{\frac{2}{3}} + 4) ({(x + 1)}^{\frac{2}{3}} - 4)) = 0$

Step 6

Factor.

Simplify.

Rewrite ${(x + 1)}^{\frac{2}{3}}$ as ${({(x + 1)}^{\frac{1}{3}})}^{2}$ .

$3 (({(x + 1)}^{\frac{2}{3}} + 4) ({({(x + 1)}^{\frac{1}{3}})}^{2} - 4)) = 0$

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

$3 (({(x + 1)}^{\frac{2}{3}} + 4) ({({(x + 1)}^{\frac{1}{3}})}^{2} - 2^{2})) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x + 1)}^{\frac{1}{3}}$ and $b = 2$ .

$3 (({(x + 1)}^{\frac{2}{3}} + 4) (({(x + 1)}^{\frac{1}{3}} + 2) ({(x + 1)}^{\frac{1}{3}} - 2))) = 0$

Remove unnecessary parentheses.

$3 (({(x + 1)}^{\frac{2}{3}} + 4) ({(x + 1)}^{\frac{1}{3}} + 2) ({(x + 1)}^{\frac{1}{3}} - 2)) = 0$

Remove unnecessary parentheses.

$3 ({(x + 1)}^{\frac{2}{3}} + 4) ({(x + 1)}^{\frac{1}{3}} + 2) ({(x + 1)}^{\frac{1}{3}} - 2) = 0$

Step 7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 1)}^{\frac{2}{3}} + 4 = 0$

${(x + 1)}^{\frac{1}{3}} + 2 = 0$

${(x + 1)}^{\frac{1}{3}} - 2 = 0$

Step 8

Set ${(x + 1)}^{\frac{2}{3}} + 4$ equal to $0$ and solve for $x$ .

Set ${(x + 1)}^{\frac{2}{3}} + 4$ equal to $0$ .

${(x + 1)}^{\frac{2}{3}} + 4 = 0$

Solve ${(x + 1)}^{\frac{2}{3}} + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

${(x + 1)}^{\frac{2}{3}} = - 4$

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

${({(x + 1)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(- 4)}^{\frac{3}{2}}$

Simplify the left side.

Simplify ${({(x + 1)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

Multiply the exponents in ${({(x + 1)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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

${(x + 1)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(- 4)}^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 1)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(- 4)}^{\frac{3}{2}}$

Rewrite the expression.

${(x + 1)}^{\frac{1}{3} \cdot 3} = \pm {(- 4)}^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x + 1)}^{\frac{1}{3} \cdot 3} = \pm {(- 4)}^{\frac{3}{2}}$

Rewrite the expression.

${(x + 1)}^{1} = \pm {(- 4)}^{\frac{3}{2}}$

Simplify.

$x + 1 = \pm {(- 4)}^{\frac{3}{2}}$

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.

$x + 1 = {(- 4)}^{\frac{3}{2}}$

Subtract $1$ from both sides of the equation.

$x = {(- 4)}^{\frac{3}{2}} - 1$

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

$x + 1 = - {(- 4)}^{\frac{3}{2}}$

Subtract $1$ from both sides of the equation.

$x = - {(- 4)}^{\frac{3}{2}} - 1$

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

$x = {(- 4)}^{\frac{3}{2}} - 1, - {(- 4)}^{\frac{3}{2}} - 1$

Step 9

Set ${(x + 1)}^{\frac{1}{3}} + 2$ equal to $0$ and solve for $x$ .

Set ${(x + 1)}^{\frac{1}{3}} + 2$ equal to $0$ .

${(x + 1)}^{\frac{1}{3}} + 2 = 0$

Solve ${(x + 1)}^{\frac{1}{3}} + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

${(x + 1)}^{\frac{1}{3}} = - 2$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(x + 1)}^{\frac{1}{3}})}^{3} = {(- 2)}^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x + 1)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(x + 1)}^{\frac{1}{3}})}^{3}$ .

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

${(x + 1)}^{1} = {(- 2)}^{3}$

Simplify.

$x + 1 = {(- 2)}^{3}$

Simplify the right side.

Raise $- 2$ to the power of $3$ .

$x + 1 = - 8$

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

Subtract $1$ from both sides of the equation.

$x = - 8 - 1$

Subtract $1$ from $- 8$ .

$x = - 9$

Step 10

Set ${(x + 1)}^{\frac{1}{3}} - 2$ equal to $0$ and solve for $x$ .

Set ${(x + 1)}^{\frac{1}{3}} - 2$ equal to $0$ .

${(x + 1)}^{\frac{1}{3}} - 2 = 0$

Solve ${(x + 1)}^{\frac{1}{3}} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

${(x + 1)}^{\frac{1}{3}} = 2$

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(x + 1)}^{\frac{1}{3}})}^{3} = 2^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x + 1)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(x + 1)}^{\frac{1}{3}})}^{3}$ .

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

${(x + 1)}^{\frac{1}{3} \cdot 3} = 2^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x + 1)}^{\frac{1}{3} \cdot 3} = 2^{3}$

Rewrite the expression.

${(x + 1)}^{1} = 2^{3}$

Simplify.

$x + 1 = 2^{3}$

Simplify the right side.

Raise $2$ to the power of $3$ .

$x + 1 = 8$

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

Subtract $1$ from both sides of the equation.

$x = 8 - 1$

Subtract $1$ from $8$ .

$x = 7$

Step 11

The final solution is all the values that make $3 ({(x + 1)}^{\frac{2}{3}} + 4) ({(x + 1)}^{\frac{1}{3}} + 2) ({(x + 1)}^{\frac{1}{3}} - 2) = 0$ true.

$x = {(- 4)}^{\frac{3}{2}} - 1, - {(- 4)}^{\frac{3}{2}} - 1, - 9, 7$

Step 12

Exclude the solutions that do not make $3 ({(x + 1)}^{\frac{2}{3}} + 4) ({(x + 1)}^{\frac{1}{3}} + 2) ({(x + 1)}^{\frac{1}{3}} - 2) = 0$ true.

$x = - 9, 7$

Solve by Factoring 3(x+1)^(4/3)=48

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