Algebra Examples

$3 \sqrt[5]{{(x + 2)}^{3}} + 3 = 27$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(x + 2)}^{3}}$ as ${(x + 2)}^{\frac{3}{5}}$ .

$3 {(x + 2)}^{\frac{3}{5}} + 3 = 27$

Step 2

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

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Step 2.1

Subtract $3$ from both sides of the equation.

$3 {(x + 2)}^{\frac{3}{5}} = 27 - 3$

Step 2.2

Subtract $3$ from $27$ .

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

Step 3

Raise each side of the equation to the power of $\frac{5}{3}$ to eliminate the fractional exponent on the left side.

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

Step 4

Simplify the left side.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

Cancel the common factor of $3$ .

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Step 4.1.2.2.1

Cancel the common factor.

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

Step 4.1.2.2.2

Rewrite the expression.

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

Step 4.1.2.3

Cancel the common factor of $5$ .

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Step 4.1.2.3.1

Cancel the common factor.

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

Step 4.1.2.3.2

Rewrite the expression.

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

Step 4.1.3

Simplify.

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

Step 4.1.4

Apply the distributive property.

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

Step 4.1.5

Simplify the expression.

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Step 4.1.5.1

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

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

Step 4.1.5.2

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

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

Step 5

Solve for $x$ .

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor.

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

Step 5.2.2.2

Divide $x$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

Divide $24$ by $3$ .

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

Step 5.2.3.1.3

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

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

Step 5.2.3.1.4

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

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

Step 5.2.3.1.5

Cancel the common factor of $3$ .

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Step 5.2.3.1.5.1

Cancel the common factor.

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

Step 5.2.3.1.5.2

Rewrite the expression.

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

Step 5.2.3.1.6

Raise $2$ to the power of $5$ .

$x = 32 + \frac{- 2 \cdot 3^{\frac{5}{3}}}{3^{\frac{5}{3}}}$

Step 5.2.3.1.7

Cancel the common factor.

$x = 32 + \frac{- 2 \cdot 3^{\frac{5}{3}}}{3^{\frac{5}{3}}}$

Step 5.2.3.1.8

Divide $- 2$ by $1$ .

$x = 32 - 2$

Step 5.2.3.2

Subtract $2$ from $32$ .

$x = 30$