Algebra Examples

$3 x^{\frac{4}{3}} + 5 = 53$

Step 1

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

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Step 1.1

Subtract $5$ from both sides of the equation.

$3 x^{\frac{4}{3}} - 53 = - 5$

Step 1.2

Add $53$ to both sides of the equation.

$3 x^{\frac{4}{3}} = - 5 + 53$

Step 1.3

Add $- 5$ and $53$ .

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

Step 2

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

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

Step 3

Simplify the left side.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Cancel the common factor of $4$ .

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Step 3.1.2.2.1

Cancel the common factor.

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

Step 3.1.2.2.2

Rewrite the expression.

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

Step 3.1.2.3

Cancel the common factor of $3$ .

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Step 3.1.2.3.1

Cancel the common factor.

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

Step 3.1.2.3.2

Rewrite the expression.

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

Step 3.1.3

Simplify.

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

Step 3.1.4

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

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

Step 4

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

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

Divide $x$ by $1$ .

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

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

Step 4.2.3.2

Simplify the expression.

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Step 4.2.3.2.1

Divide $48$ by $3$ .

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

Step 4.2.3.2.2

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

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

Step 4.2.3.2.3

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

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

Step 4.2.3.3

Cancel the common factor of $4$ .

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Step 4.2.3.3.1

Cancel the common factor.

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

Step 4.2.3.3.2

Rewrite the expression.

$x = 2^{3}$

Step 4.2.3.4

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

$x = 8$

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor.

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

Step 4.4.2.2

Divide $x$ by $1$ .

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

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Move the negative in front of the fraction.

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

Step 4.4.3.2

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

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

Step 4.4.3.3

Simplify the expression.

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Step 4.4.3.3.1

Divide $48$ by $3$ .

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

Step 4.4.3.3.2

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

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

Step 4.4.3.3.3

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

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

Step 4.4.3.4

Cancel the common factor of $4$ .

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Step 4.4.3.4.1

Cancel the common factor.

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

Step 4.4.3.4.2

Rewrite the expression.

$x = - 2^{3}$

Step 4.4.3.5

Simplify the expression.

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Step 4.4.3.5.1

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

$x = - 1 \cdot 8$

Step 4.4.3.5.2

Multiply $- 1$ by $8$ .

$x = - 8$

Step 4.5

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

$x = 8, - 8$