Solve for x 3(x-5)^(3/2)-6=18

$3 {(x - 5)}^{\frac{3}{2}} - 6 = 18$

Step 1

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

Add $6$ to both sides of the equation.

$3 {(x - 5)}^{\frac{3}{2}} = 18 + 6$

Add $18$ and $6$ .

$3 {(x - 5)}^{\frac{3}{2}} = 24$

Step 2

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

${(3 {(x - 5)}^{\frac{3}{2}})}^{\frac{2}{3}} = 24^{\frac{2}{3}}$

Step 3

Simplify the left side.

Simplify ${(3 {(x - 5)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Apply the product rule to $3 {(x - 5)}^{\frac{3}{2}}$ .

$3^{\frac{2}{3}} {({(x - 5)}^{\frac{3}{2}})}^{\frac{2}{3}} = 24^{\frac{2}{3}}$

Multiply the exponents in ${({(x - 5)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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

$3^{\frac{2}{3}} {(x - 5)}^{\frac{3}{2} \cdot \frac{2}{3}} = 24^{\frac{2}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$3^{\frac{2}{3}} {(x - 5)}^{\frac{3}{2} \cdot \frac{2}{3}} = 24^{\frac{2}{3}}$

Rewrite the expression.

$3^{\frac{2}{3}} {(x - 5)}^{\frac{1}{2} \cdot 2} = 24^{\frac{2}{3}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3^{\frac{2}{3}} {(x - 5)}^{\frac{1}{2} \cdot 2} = 24^{\frac{2}{3}}$

Rewrite the expression.

$3^{\frac{2}{3}} {(x - 5)}^{1} = 24^{\frac{2}{3}}$

Simplify.

$3^{\frac{2}{3}} (x - 5) = 24^{\frac{2}{3}}$

Apply the distributive property.

$3^{\frac{2}{3}} x + 3^{\frac{2}{3}} \cdot - 5 = 24^{\frac{2}{3}}$

Simplify the expression.

Move $- 5$ to the left of $3^{\frac{2}{3}}$ .

$3^{\frac{2}{3}} x - 5 \cdot 3^{\frac{2}{3}} = 24^{\frac{2}{3}}$

Reorder factors in $3^{\frac{2}{3}} x - 5 \cdot 3^{\frac{2}{3}}$ .

$x \cdot 3^{\frac{2}{3}} - 5 \cdot 3^{\frac{2}{3}} = 24^{\frac{2}{3}}$

Step 4

Solve for $x$ .

Add $5 \cdot 3^{\frac{2}{3}}$ to both sides of the equation.

$x \cdot 3^{\frac{2}{3}} = 24^{\frac{2}{3}} + 5 \cdot 3^{\frac{2}{3}}$

Divide each term in $(x \cdot 3^{\frac{2}{3}}) = 24^{\frac{2}{3}} + 5 \cdot 3^{\frac{2}{3}}$ by $3^{\frac{2}{3}}$ and simplify.

Divide each term in $(x \cdot 3^{\frac{2}{3}}) = 24^{\frac{2}{3}} + 5 \cdot 3^{\frac{2}{3}}$ by $3^{\frac{2}{3}}$ .

$\frac{(x \cdot 3^{\frac{2}{3}})}{3^{\frac{2}{3}}} = \frac{24^{\frac{2}{3}}}{3^{\frac{2}{3}}} + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Simplify the left side.

Cancel the common factor.

$\frac{x \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}} = \frac{24^{\frac{2}{3}}}{3^{\frac{2}{3}}} + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Divide $x$ by $1$ .

$x = \frac{24^{\frac{2}{3}}}{3^{\frac{2}{3}}} + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Simplify the right side.

Simplify each term.

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

$x = {(\frac{24}{3})}^{\frac{2}{3}} + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Divide $24$ by $3$ .

$x = 8^{\frac{2}{3}} + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Rewrite $8$ as $2^{3}$ .

$x = {(2^{3})}^{\frac{2}{3}} + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

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

$x = 2^{3 (\frac{2}{3})} + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x = 2^{3 (\frac{2}{3})} + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Rewrite the expression.

$x = 2^{2} + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

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

$x = 4 + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Cancel the common factor.

$x = 4 + \frac{5 \cdot 3^{\frac{2}{3}}}{3^{\frac{2}{3}}}$

Divide $5$ by $1$ .

$x = 4 + 5$

Add $4$ and $5$ .

$x = 9$

Solve for x 3(x-5)^(3/2)-6=18

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