Algebra Examples

$\frac{1}{2} {(5 x + 1)}^{\frac{1}{3}} + \frac{5}{2} = 4$

Step 1

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

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Step 1.1

Subtract $\frac{5}{2}$ from both sides of the equation.

$\frac{1}{2} \cdot {(5 x + 1)}^{\frac{1}{3}} = 4 - \frac{5}{2}$

Step 1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} \cdot {(5 x + 1)}^{\frac{1}{3}} = 4 \cdot \frac{2}{2} - \frac{5}{2}$

Step 1.3

Combine $4$ and $\frac{2}{2}$ .

$\frac{1}{2} \cdot {(5 x + 1)}^{\frac{1}{3}} = \frac{4 \cdot 2}{2} - \frac{5}{2}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot {(5 x + 1)}^{\frac{1}{3}} = \frac{4 \cdot 2 - 5}{2}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $4$ by $2$ .

$\frac{1}{2} \cdot {(5 x + 1)}^{\frac{1}{3}} = \frac{8 - 5}{2}$

Step 1.5.2

Subtract $5$ from $8$ .

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

Step 2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Step 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify ${(\frac{1}{2} \cdot {(5 x + 1)}^{\frac{1}{3}})}^{3}$ .

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Step 3.1.1.1

Combine fractions.

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Step 3.1.1.1.1

Combine $\frac{1}{2}$ and ${(5 x + 1)}^{\frac{1}{3}}$ .

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

Step 3.1.1.1.2

Apply the product rule to $\frac{{(5 x + 1)}^{\frac{1}{3}}}{2}$ .

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

Step 3.1.1.2

Simplify the numerator.

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Step 3.1.1.2.1

Multiply the exponents in ${({(5 x + 1)}^{\frac{1}{3}})}^{3}$ .

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Step 3.1.1.2.1.1

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

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

Step 3.1.1.2.1.2

Cancel the common factor of $3$ .

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Step 3.1.1.2.1.2.1

Cancel the common factor.

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

Step 3.1.1.2.1.2.2

Rewrite the expression.

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

Step 3.1.1.2.2

Simplify.

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

Step 3.1.1.3

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

$\frac{5 x + 1}{8} = {(\frac{3}{2})}^{3}$

Step 3.1.1.4

Split the fraction $\frac{5 x + 1}{8}$ into two fractions.

$\frac{5 x}{8} + \frac{1}{8} = {(\frac{3}{2})}^{3}$

Step 3.2

Simplify the right side.

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Step 3.2.1

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

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Step 3.2.1.1

Apply the product rule to $\frac{3}{2}$ .

$\frac{5 x}{8} + \frac{1}{8} = \frac{3^{3}}{2^{3}}$

Step 3.2.1.2

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

$\frac{5 x}{8} + \frac{1}{8} = \frac{27}{2^{3}}$

Step 3.2.1.3

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

$\frac{5 x}{8} + \frac{1}{8} = \frac{27}{8}$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $\frac{1}{8}$ from both sides of the equation.

$\frac{5 x}{8} = \frac{27}{8} - \frac{1}{8}$

Step 4.1.2

Combine the numerators over the common denominator.

$\frac{5 x}{8} = \frac{27 - 1}{8}$

Step 4.1.3

Subtract $1$ from $27$ .

$\frac{5 x}{8} = \frac{26}{8}$

Step 4.1.4

Cancel the common factor of $26$ and $8$ .

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Step 4.1.4.1

Factor $2$ out of $26$ .

$\frac{5 x}{8} = \frac{2 (13)}{8}$

Step 4.1.4.2

Cancel the common factors.

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Step 4.1.4.2.1

Factor $2$ out of $8$ .

$\frac{5 x}{8} = \frac{2 \cdot 13}{2 \cdot 4}$

Step 4.1.4.2.2

Cancel the common factor.

$\frac{5 x}{8} = \frac{2 \cdot 13}{2 \cdot 4}$

Step 4.1.4.2.3

Rewrite the expression.

$\frac{5 x}{8} = \frac{13}{4}$

Step 4.2

Multiply both sides of the equation by $\frac{8}{5}$ .

$\frac{8}{5} \cdot \frac{5 x}{8} = \frac{8}{5} \cdot \frac{13}{4}$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Simplify $\frac{8}{5} \cdot \frac{5 x}{8}$ .

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Step 4.3.1.1.1

Cancel the common factor of $8$ .

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Step 4.3.1.1.1.1

Cancel the common factor.

$\frac{8}{5} \cdot \frac{5 x}{8} = \frac{8}{5} \cdot \frac{13}{4}$

Step 4.3.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{8}{5} \cdot \frac{13}{4}$

Step 4.3.1.1.2

Cancel the common factor of $5$ .

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Step 4.3.1.1.2.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{8}{5} \cdot \frac{13}{4}$

Step 4.3.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{8}{5} \cdot \frac{13}{4}$

Step 4.3.1.1.2.3

Rewrite the expression.

$x = \frac{8}{5} \cdot \frac{13}{4}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Simplify $\frac{8}{5} \cdot \frac{13}{4}$ .

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Step 4.3.2.1.1

Cancel the common factor of $4$ .

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Step 4.3.2.1.1.1

Factor $4$ out of $8$ .

$x = \frac{4 (2)}{5} \cdot \frac{13}{4}$

Step 4.3.2.1.1.2

Cancel the common factor.

$x = \frac{4 \cdot 2}{5} \cdot \frac{13}{4}$

Step 4.3.2.1.1.3

Rewrite the expression.

$x = \frac{2}{5} \cdot 13$

Step 4.3.2.1.2

Combine $\frac{2}{5}$ and $13$ .

$x = \frac{2 \cdot 13}{5}$

Step 4.3.2.1.3

Multiply $2$ by $13$ .

$x = \frac{26}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{26}{5}$

Decimal Form:

$x = 5.2$

Mixed Number Form:

$x = 5 \frac{1}{5}$