Solve for x -646=-3(65-x)^(3/2)+2

$- 646 = - 3 {(65 - x)}^{\frac{3}{2}} + 2$

Step 1

Rewrite the equation as $- 3 {(65 - x)}^{\frac{3}{2}} + 2 = - 646$ .

$- 3 {(65 - x)}^{\frac{3}{2}} + 2 = - 646$

Step 2

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

Subtract $2$ from both sides of the equation.

$- 3 {(65 - x)}^{\frac{3}{2}} = - 646 - 2$

Subtract $2$ from $- 646$ .

$- 3 {(65 - x)}^{\frac{3}{2}} = - 648$

Step 3

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

${(- 3 {(65 - x)}^{\frac{3}{2}})}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}}$

Step 4

Simplify the left side.

Simplify ${(- 3 {(65 - x)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Apply the product rule to $- 3 {(65 - x)}^{\frac{3}{2}}$ .

${(- 3)}^{\frac{2}{3}} {({(65 - x)}^{\frac{3}{2}})}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}}$

Multiply the exponents in ${({(65 - x)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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

${(- 3)}^{\frac{2}{3}} {(65 - x)}^{\frac{3}{2} \cdot \frac{2}{3}} = {(- 648)}^{\frac{2}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(- 3)}^{\frac{2}{3}} {(65 - x)}^{\frac{3}{2} \cdot \frac{2}{3}} = {(- 648)}^{\frac{2}{3}}$

Rewrite the expression.

${(- 3)}^{\frac{2}{3}} {(65 - x)}^{\frac{1}{2} \cdot 2} = {(- 648)}^{\frac{2}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- 3)}^{\frac{2}{3}} {(65 - x)}^{\frac{1}{2} \cdot 2} = {(- 648)}^{\frac{2}{3}}$

Rewrite the expression.

${(- 3)}^{\frac{2}{3}} {(65 - x)}^{1} = {(- 648)}^{\frac{2}{3}}$

Simplify.

${(- 3)}^{\frac{2}{3}} (65 - x) = {(- 648)}^{\frac{2}{3}}$

Apply the distributive property.

${(- 3)}^{\frac{2}{3}} \cdot 65 + {(- 3)}^{\frac{2}{3}} (- x) = {(- 648)}^{\frac{2}{3}}$

Reorder.

Move $65$ to the left of ${(- 3)}^{\frac{2}{3}}$ .

$65 \cdot {(- 3)}^{\frac{2}{3}} + {(- 3)}^{\frac{2}{3}} (- x) = {(- 648)}^{\frac{2}{3}}$

Reorder factors in $65 {(- 3)}^{\frac{2}{3}} + {(- 3)}^{\frac{2}{3}} (- x)$ .

$65 {(- 3)}^{\frac{2}{3}} - x {(- 3)}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}}$

Step 5

Solve for $x$ .

Subtract $65 {(- 3)}^{\frac{2}{3}}$ from both sides of the equation.

$- x {(- 3)}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}} - 65 {(- 3)}^{\frac{2}{3}}$

Divide each term in $- x {(- 3)}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}} - 65 {(- 3)}^{\frac{2}{3}}$ by $- {(- 3)}^{\frac{2}{3}}$ and simplify.

Divide each term in $- x {(- 3)}^{\frac{2}{3}} = {(- 648)}^{\frac{2}{3}} - 65 {(- 3)}^{\frac{2}{3}}$ by $- {(- 3)}^{\frac{2}{3}}$ .

$\frac{- x {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}} = \frac{{(- 648)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x {(- 3)}^{\frac{2}{3}}}{{(- 3)}^{\frac{2}{3}}} = \frac{{(- 648)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Cancel the common factor.

$\frac{x {(- 3)}^{\frac{2}{3}}}{{(- 3)}^{\frac{2}{3}}} = \frac{{(- 648)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Divide $x$ by $1$ .

$x = \frac{{(- 648)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{{(- 648)}^{\frac{2}{3}}}{{(- 3)}^{\frac{2}{3}}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

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

$x = - {(\frac{- 648}{- 3})}^{\frac{2}{3}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Divide $- 648$ by $- 3$ .

$x = - 216^{\frac{2}{3}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

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

$x = - {(6^{3})}^{\frac{2}{3}} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

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

$x = - 6^{3 (\frac{2}{3})} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x = - 6^{3 (\frac{2}{3})} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Rewrite the expression.

$x = - 6^{2} + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Raise $6$ to the power of $2$ .

$x = - 1 \cdot 36 + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Multiply $- 1$ by $36$ .

$x = - 36 + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Cancel the common factor.

$x = - 36 + \frac{- 65 {(- 3)}^{\frac{2}{3}}}{- {(- 3)}^{\frac{2}{3}}}$

Rewrite the expression.

$x = - 36 + \frac{- 65}{- 1}$

Move the negative one from the denominator of $\frac{- 65}{- 1}$ .

$x = - 36 - 1 \cdot - 65$

Rewrite $- 1 \cdot - 65$ as $- - 65$ .

$x = - 36 - - 65$

Multiply $- 1$ by $- 65$ .

$x = - 36 + 65$

Add $- 36$ and $65$ .

$x = 29$

Solve for x -646=-3(65-x)^(3/2)+2

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