Find the Roots (Zeros) 3(x-4)^(4/3)+16=64

$3 {(x - 4)}^{\frac{4}{3}} + 16 = 64$

Step 1

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

Subtract $16$ from both sides of the equation.

$3 {(x - 4)}^{\frac{4}{3}} = 64 - 16$

Subtract $16$ from $64$ .

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

Step 2

Raise each side of the equation to the power of $\frac{3}{4}$ to eliminate the fractional exponent on the left side.

${(3 {(x - 4)}^{\frac{4}{3}})}^{\frac{3}{4}} = \pm 48^{\frac{3}{4}}$

Step 3

Simplify the left side.

Simplify ${(3 {(x - 4)}^{\frac{4}{3}})}^{\frac{3}{4}}$ .

Apply the product rule to $3 {(x - 4)}^{\frac{4}{3}}$ .

$3^{\frac{3}{4}} {({(x - 4)}^{\frac{4}{3}})}^{\frac{3}{4}} = \pm 48^{\frac{3}{4}}$

Multiply the exponents in ${({(x - 4)}^{\frac{4}{3}})}^{\frac{3}{4}}$ .

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

$3^{\frac{3}{4}} {(x - 4)}^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 48^{\frac{3}{4}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$3^{\frac{3}{4}} {(x - 4)}^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 48^{\frac{3}{4}}$

Rewrite the expression.

$3^{\frac{3}{4}} {(x - 4)}^{\frac{1}{3} \cdot 3} = \pm 48^{\frac{3}{4}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$3^{\frac{3}{4}} {(x - 4)}^{\frac{1}{3} \cdot 3} = \pm 48^{\frac{3}{4}}$

Rewrite the expression.

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

Simplify.

$3^{\frac{3}{4}} (x - 4) = \pm 48^{\frac{3}{4}}$

Apply the distributive property.

$3^{\frac{3}{4}} x + 3^{\frac{3}{4}} \cdot - 4 = \pm 48^{\frac{3}{4}}$

Simplify the expression.

Move $- 4$ to the left of $3^{\frac{3}{4}}$ .

$3^{\frac{3}{4}} x - 4 \cdot 3^{\frac{3}{4}} = \pm 48^{\frac{3}{4}}$

Reorder factors in $3^{\frac{3}{4}} x - 4 \cdot 3^{\frac{3}{4}}$ .

$x \cdot 3^{\frac{3}{4}} - 4 \cdot 3^{\frac{3}{4}} = \pm 48^{\frac{3}{4}}$

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.

$x \cdot 3^{\frac{3}{4}} - 4 \cdot 3^{\frac{3}{4}} = 48^{\frac{3}{4}}$

Add $4 \cdot 3^{\frac{3}{4}}$ to both sides of the equation.

$x \cdot 3^{\frac{3}{4}} = 48^{\frac{3}{4}} + 4 \cdot 3^{\frac{3}{4}}$

Divide each term in $(x \cdot 3^{\frac{3}{4}}) = 48^{\frac{3}{4}} + 4 \cdot 3^{\frac{3}{4}}$ by $3^{\frac{3}{4}}$ and simplify.

Divide each term in $(x \cdot 3^{\frac{3}{4}}) = 48^{\frac{3}{4}} + 4 \cdot 3^{\frac{3}{4}}$ by $3^{\frac{3}{4}}$ .

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

Simplify the left side.

Cancel the common factor.

$\frac{x \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}} = \frac{48^{\frac{3}{4}}}{3^{\frac{3}{4}}} + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

Divide $x$ by $1$ .

$x = \frac{48^{\frac{3}{4}}}{3^{\frac{3}{4}}} + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

Simplify the right side.

Simplify each term.

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

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

Divide $48$ by $3$ .

$x = 16^{\frac{3}{4}} + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

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

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$x = 8 + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

Cancel the common factor.

$x = 8 + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

Divide $4$ by $1$ .

$x = 8 + 4$

Add $8$ and $4$ .

$x = 12$

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

$x \cdot 3^{\frac{3}{4}} - 4 \cdot 3^{\frac{3}{4}} = - 48^{\frac{3}{4}}$

Add $4 \cdot 3^{\frac{3}{4}}$ to both sides of the equation.

$x \cdot 3^{\frac{3}{4}} = - 48^{\frac{3}{4}} + 4 \cdot 3^{\frac{3}{4}}$

Divide each term in $(x \cdot 3^{\frac{3}{4}}) = - 48^{\frac{3}{4}} + 4 \cdot 3^{\frac{3}{4}}$ by $3^{\frac{3}{4}}$ and simplify.

Divide each term in $(x \cdot 3^{\frac{3}{4}}) = - 48^{\frac{3}{4}} + 4 \cdot 3^{\frac{3}{4}}$ by $3^{\frac{3}{4}}$ .

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

Simplify the left side.

Cancel the common factor.

$\frac{x \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}} = \frac{- 48^{\frac{3}{4}}}{3^{\frac{3}{4}}} + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

Divide $x$ by $1$ .

$x = \frac{- 48^{\frac{3}{4}}}{3^{\frac{3}{4}}} + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{48^{\frac{3}{4}}}{3^{\frac{3}{4}}} + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

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

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

Divide $48$ by $3$ .

$x = - 16^{\frac{3}{4}} + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

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

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

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

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

$x = - 1 \cdot 8 + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

Multiply $- 1$ by $8$ .

$x = - 8 + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

Cancel the common factor.

$x = - 8 + \frac{4 \cdot 3^{\frac{3}{4}}}{3^{\frac{3}{4}}}$

Divide $4$ by $1$ .

$x = - 8 + 4$

Add $- 8$ and $4$ .

$x = - 4$

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

$x = 12, - 4$

Step 5

Find the Roots (Zeros) 3(x-4)^(4/3)+16=64

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