Algebra Examples

${(2 x + 3)}^{\frac{4}{3}} - 3 = 13$

Step 1

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

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Step 1.1

Add $3$ to both sides of the equation.

${(2 x + 3)}^{\frac{4}{3}} = 13 + 3$

Step 1.2

Add $13$ and $3$ .

${(2 x + 3)}^{\frac{4}{3}} = 16$

Step 2

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

${({(2 x + 3)}^{\frac{4}{3}})}^{\frac{3}{4}} = \pm 16^{\frac{3}{4}}$

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

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Step 3.1.1.1

Multiply the exponents in ${({(2 x + 3)}^{\frac{4}{3}})}^{\frac{3}{4}}$ .

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Step 3.1.1.1.1

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

${(2 x + 3)}^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 16^{\frac{3}{4}}$

Step 3.1.1.1.2

Cancel the common factor of $4$ .

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Step 3.1.1.1.2.1

Cancel the common factor.

${(2 x + 3)}^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 16^{\frac{3}{4}}$

Step 3.1.1.1.2.2

Rewrite the expression.

${(2 x + 3)}^{\frac{1}{3} \cdot 3} = \pm 16^{\frac{3}{4}}$

Step 3.1.1.1.3

Cancel the common factor of $3$ .

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Step 3.1.1.1.3.1

Cancel the common factor.

${(2 x + 3)}^{\frac{1}{3} \cdot 3} = \pm 16^{\frac{3}{4}}$

Step 3.1.1.1.3.2

Rewrite the expression.

${(2 x + 3)}^{1} = \pm 16^{\frac{3}{4}}$

Step 3.1.1.2

Simplify.

$2 x + 3 = \pm 16^{\frac{3}{4}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\pm 16^{\frac{3}{4}}$ .

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Step 3.2.1.1

Simplify the expression.

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

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

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

Step 3.2.1.2

Cancel the common factor of $4$ .

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Step 3.2.1.2.1

Cancel the common factor.

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

Step 3.2.1.2.2

Rewrite the expression.

$2 x + 3 = \pm 2^{3}$

Step 3.2.1.3

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

$2 x + 3 = \pm 8$

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.

$2 x + 3 = 8$

Step 4.2

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

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Step 4.2.1

Subtract $3$ from both sides of the equation.

$2 x = 8 - 3$

Step 4.2.2

Subtract $3$ from $8$ .

$2 x = 5$

Step 4.3

Divide each term in $2 x = 5$ by $2$ and simplify.

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Step 4.3.1

Divide each term in $2 x = 5$ by $2$ .

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

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.1

Cancel the common factor.

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

Step 4.3.2.1.2

Divide $x$ by $1$ .

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

Step 4.4

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

$2 x + 3 = - 8$

Step 4.5

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

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Step 4.5.1

Subtract $3$ from both sides of the equation.

$2 x = - 8 - 3$

Step 4.5.2

Subtract $3$ from $- 8$ .

$2 x = - 11$

Step 4.6

Divide each term in $2 x = - 11$ by $2$ and simplify.

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Step 4.6.1

Divide each term in $2 x = - 11$ by $2$ .

$\frac{2 x}{2} = \frac{- 11}{2}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $2$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 11}{2}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{2}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Move the negative in front of the fraction.

$x = - \frac{11}{2}$

Step 4.7

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

$x = \frac{5}{2}, - \frac{11}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{2}, - \frac{11}{2}$

Decimal Form:

$x = 2.5, - 5.5$

Mixed Number Form:

$x = 2 \frac{1}{2}, - 5 \frac{1}{2}$