Solve for x 1+2(x+3)^(2/3)=9

$1 + 2 {(x + 3)}^{\frac{2}{3}} = 9$

Step 1

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

Subtract $1$ from both sides of the equation.

$2 {(x + 3)}^{\frac{2}{3}} = 9 - 1$

Subtract $1$ from $9$ .

$2 {(x + 3)}^{\frac{2}{3}} = 8$

Step 2

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

${(2 {(x + 3)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 8^{\frac{3}{2}}$

Step 3

Simplify the left side.

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

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

$2^{\frac{3}{2}} {({(x + 3)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 8^{\frac{3}{2}}$

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

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

$2^{\frac{3}{2}} {(x + 3)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 8^{\frac{3}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2^{\frac{3}{2}} {(x + 3)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 8^{\frac{3}{2}}$

Rewrite the expression.

$2^{\frac{3}{2}} {(x + 3)}^{\frac{1}{3} \cdot 3} = \pm 8^{\frac{3}{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$2^{\frac{3}{2}} {(x + 3)}^{\frac{1}{3} \cdot 3} = \pm 8^{\frac{3}{2}}$

Rewrite the expression.

$2^{\frac{3}{2}} {(x + 3)}^{1} = \pm 8^{\frac{3}{2}}$

Simplify.

$2^{\frac{3}{2}} (x + 3) = \pm 8^{\frac{3}{2}}$

Apply the distributive property.

$2^{\frac{3}{2}} x + 2^{\frac{3}{2}} \cdot 3 = \pm 8^{\frac{3}{2}}$

Simplify the expression.

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

$2^{\frac{3}{2}} x + 3 \cdot 2^{\frac{3}{2}} = \pm 8^{\frac{3}{2}}$

Reorder factors in $2^{\frac{3}{2}} x + 3 \cdot 2^{\frac{3}{2}}$ .

$x \cdot 2^{\frac{3}{2}} + 3 \cdot 2^{\frac{3}{2}} = \pm 8^{\frac{3}{2}}$

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.

$x \cdot 2^{\frac{3}{2}} + 3 \cdot 2^{\frac{3}{2}} = 8^{\frac{3}{2}}$

Subtract $3 \cdot 2^{\frac{3}{2}}$ from both sides of the equation.

$x \cdot 2^{\frac{3}{2}} = 8^{\frac{3}{2}} - 3 \cdot 2^{\frac{3}{2}}$

Divide each term in $(x \cdot 2^{\frac{3}{2}}) = 8^{\frac{3}{2}} - 3 \cdot 2^{\frac{3}{2}}$ by $2^{\frac{3}{2}}$ and simplify.

Divide each term in $(x \cdot 2^{\frac{3}{2}}) = 8^{\frac{3}{2}} - 3 \cdot 2^{\frac{3}{2}}$ by $2^{\frac{3}{2}}$ .

$\frac{(x \cdot 2^{\frac{3}{2}})}{2^{\frac{3}{2}}} = \frac{8^{\frac{3}{2}}}{2^{\frac{3}{2}}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Simplify the left side.

Cancel the common factor.

$\frac{x \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}} = \frac{8^{\frac{3}{2}}}{2^{\frac{3}{2}}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Divide $x$ by $1$ .

$x = \frac{8^{\frac{3}{2}}}{2^{\frac{3}{2}}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Simplify the right side.

Simplify each term.

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

$x = {(\frac{8}{2})}^{\frac{3}{2}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Divide $8$ by $2$ .

$x = 4^{\frac{3}{2}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$x = 8 + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Cancel the common factor.

$x = 8 + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Divide $- 3$ by $1$ .

$x = 8 - 3$

Subtract $3$ from $8$ .

$x = 5$

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

$x \cdot 2^{\frac{3}{2}} + 3 \cdot 2^{\frac{3}{2}} = - 8^{\frac{3}{2}}$

Subtract $3 \cdot 2^{\frac{3}{2}}$ from both sides of the equation.

$x \cdot 2^{\frac{3}{2}} = - 8^{\frac{3}{2}} - 3 \cdot 2^{\frac{3}{2}}$

Divide each term in $(x \cdot 2^{\frac{3}{2}}) = - 8^{\frac{3}{2}} - 3 \cdot 2^{\frac{3}{2}}$ by $2^{\frac{3}{2}}$ and simplify.

Divide each term in $(x \cdot 2^{\frac{3}{2}}) = - 8^{\frac{3}{2}} - 3 \cdot 2^{\frac{3}{2}}$ by $2^{\frac{3}{2}}$ .

$\frac{(x \cdot 2^{\frac{3}{2}})}{2^{\frac{3}{2}}} = \frac{- 8^{\frac{3}{2}}}{2^{\frac{3}{2}}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Simplify the left side.

Cancel the common factor.

$\frac{x \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}} = \frac{- 8^{\frac{3}{2}}}{2^{\frac{3}{2}}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Divide $x$ by $1$ .

$x = \frac{- 8^{\frac{3}{2}}}{2^{\frac{3}{2}}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{8^{\frac{3}{2}}}{2^{\frac{3}{2}}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

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

$x = - {(\frac{8}{2})}^{\frac{3}{2}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Divide $8$ by $2$ .

$x = - 4^{\frac{3}{2}} + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$x = - 1 \cdot 8 + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Multiply $- 1$ by $8$ .

$x = - 8 + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Cancel the common factor.

$x = - 8 + \frac{- 3 \cdot 2^{\frac{3}{2}}}{2^{\frac{3}{2}}}$

Divide $- 3$ by $1$ .

$x = - 8 - 3$

Subtract $3$ from $- 8$ .

$x = - 11$

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

$x = 5, - 11$

Solve for x 1+2(x+3)^(2/3)=9

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