Solve by Factoring (x+6)^(3/5)=8

${(x + 6)}^{\frac{3}{5}} = 8$

Step 1

Subtract $8$ from both sides of the equation.

${(x + 6)}^{\frac{3}{5}} - 8 = 0$

Step 2

Rewrite ${(x + 6)}^{\frac{3}{5}}$ as ${({(x + 6)}^{\frac{1}{5}})}^{3}$ .

${({(x + 6)}^{\frac{1}{5}})}^{3} - 8 = 0$

Step 3

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

${({(x + 6)}^{\frac{1}{5}})}^{3} - 2^{3} = 0$

Step 4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = {(x + 6)}^{\frac{1}{5}}$ and $b = 2$ .

$({(x + 6)}^{\frac{1}{5}} - 2) ({({(x + 6)}^{\frac{1}{5}})}^{2} + {(x + 6)}^{\frac{1}{5}} \cdot 2 + 2^{2}) = 0$

Step 5

Simplify.

Multiply the exponents in ${({(x + 6)}^{\frac{1}{5}})}^{2}$ .

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

$({(x + 6)}^{\frac{1}{5}} - 2) ({(x + 6)}^{\frac{1}{5} \cdot 2} + {(x + 6)}^{\frac{1}{5}} \cdot 2 + 2^{2}) = 0$

Combine $\frac{1}{5}$ and $2$ .

$({(x + 6)}^{\frac{1}{5}} - 2) ({(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{1}{5}} \cdot 2 + 2^{2}) = 0$

Move $2$ to the left of ${(x + 6)}^{\frac{1}{5}}$ .

$({(x + 6)}^{\frac{1}{5}} - 2) ({(x + 6)}^{\frac{2}{5}} + 2 {(x + 6)}^{\frac{1}{5}} + 2^{2}) = 0$

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

$({(x + 6)}^{\frac{1}{5}} - 2) ({(x + 6)}^{\frac{2}{5}} + 2 {(x + 6)}^{\frac{1}{5}} + 4) = 0$

Reorder terms.

$({(x + 6)}^{\frac{1}{5}} - 2) (2 {(x + 6)}^{\frac{1}{5}} + {(x + 6)}^{\frac{2}{5}} + 4) = 0$

Step 6

Simplify $({(x + 6)}^{\frac{1}{5}} - 2) (2 {(x + 6)}^{\frac{1}{5}} + {(x + 6)}^{\frac{2}{5}} + 4)$ .

Expand $({(x + 6)}^{\frac{1}{5}} - 2) (2 {(x + 6)}^{\frac{1}{5}} + {(x + 6)}^{\frac{2}{5}} + 4)$ by multiplying each term in the first expression by each term in the second expression.

${(x + 6)}^{\frac{1}{5}} (2 {(x + 6)}^{\frac{1}{5}}) + {(x + 6)}^{\frac{1}{5}} {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{1}{5}} \cdot 4 - 2 (2 {(x + 6)}^{\frac{1}{5}}) - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 {(x + 6)}^{\frac{1}{5}} {(x + 6)}^{\frac{1}{5}} + {(x + 6)}^{\frac{1}{5}} {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{1}{5}} \cdot 4 - 2 (2 {(x + 6)}^{\frac{1}{5}}) - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Multiply ${(x + 6)}^{\frac{1}{5}}$ by ${(x + 6)}^{\frac{1}{5}}$ by adding the exponents.

Move ${(x + 6)}^{\frac{1}{5}}$ .

$2 ({(x + 6)}^{\frac{1}{5}} {(x + 6)}^{\frac{1}{5}}) + {(x + 6)}^{\frac{1}{5}} {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{1}{5}} \cdot 4 - 2 (2 {(x + 6)}^{\frac{1}{5}}) - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 {(x + 6)}^{\frac{1}{5} + \frac{1}{5}} + {(x + 6)}^{\frac{1}{5}} {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{1}{5}} \cdot 4 - 2 (2 {(x + 6)}^{\frac{1}{5}}) - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Combine the numerators over the common denominator.

$2 {(x + 6)}^{\frac{1 + 1}{5}} + {(x + 6)}^{\frac{1}{5}} {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{1}{5}} \cdot 4 - 2 (2 {(x + 6)}^{\frac{1}{5}}) - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Add $1$ and $1$ .

$2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{1}{5}} {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{1}{5}} \cdot 4 - 2 (2 {(x + 6)}^{\frac{1}{5}}) - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Multiply ${(x + 6)}^{\frac{1}{5}}$ by ${(x + 6)}^{\frac{2}{5}}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{1}{5} + \frac{2}{5}} + {(x + 6)}^{\frac{1}{5}} \cdot 4 - 2 (2 {(x + 6)}^{\frac{1}{5}}) - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Combine the numerators over the common denominator.

$2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{1 + 2}{5}} + {(x + 6)}^{\frac{1}{5}} \cdot 4 - 2 (2 {(x + 6)}^{\frac{1}{5}}) - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Add $1$ and $2$ .

$2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{3}{5}} + {(x + 6)}^{\frac{1}{5}} \cdot 4 - 2 (2 {(x + 6)}^{\frac{1}{5}}) - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Move $4$ to the left of ${(x + 6)}^{\frac{1}{5}}$ .

$2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{3}{5}} + 4 \cdot {(x + 6)}^{\frac{1}{5}} - 2 (2 {(x + 6)}^{\frac{1}{5}}) - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Multiply $2$ by $- 2$ .

$2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{3}{5}} + 4 {(x + 6)}^{\frac{1}{5}} - 4 {(x + 6)}^{\frac{1}{5}} - 2 {(x + 6)}^{\frac{2}{5}} - 2 \cdot 4 = 0$

Multiply $- 2$ by $4$ .

$2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{3}{5}} + 4 {(x + 6)}^{\frac{1}{5}} - 4 {(x + 6)}^{\frac{1}{5}} - 2 {(x + 6)}^{\frac{2}{5}} - 8 = 0$

Combine the opposite terms in $2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{3}{5}} + 4 {(x + 6)}^{\frac{1}{5}} - 4 {(x + 6)}^{\frac{1}{5}} - 2 {(x + 6)}^{\frac{2}{5}} - 8$ .

Subtract $4 {(x + 6)}^{\frac{1}{5}}$ from $4 {(x + 6)}^{\frac{1}{5}}$ .

$2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{3}{5}} + 0 - 2 {(x + 6)}^{\frac{2}{5}} - 8 = 0$

Add $2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{3}{5}}$ and $0$ .

$2 {(x + 6)}^{\frac{2}{5}} + {(x + 6)}^{\frac{3}{5}} - 2 {(x + 6)}^{\frac{2}{5}} - 8 = 0$

Subtract $2 {(x + 6)}^{\frac{2}{5}}$ from $2 {(x + 6)}^{\frac{2}{5}}$ .

$0 + {(x + 6)}^{\frac{3}{5}} - 8 = 0$

Add $0$ and ${(x + 6)}^{\frac{3}{5}}$ .

${(x + 6)}^{\frac{3}{5}} - 8 = 0$

Step 7

Add $8$ to both sides of the equation.

${(x + 6)}^{\frac{3}{5}} = 8$

Step 8

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

${({(x + 6)}^{\frac{3}{5}})}^{\frac{5}{3}} = 8^{\frac{5}{3}}$

Step 9

Simplify the exponent.

Simplify the left side.

Simplify ${({(x + 6)}^{\frac{3}{5}})}^{\frac{5}{3}}$ .

Multiply the exponents in ${({(x + 6)}^{\frac{3}{5}})}^{\frac{5}{3}}$ .

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

${(x + 6)}^{\frac{3}{5} \cdot \frac{5}{3}} = 8^{\frac{5}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x + 6)}^{\frac{3}{5} \cdot \frac{5}{3}} = 8^{\frac{5}{3}}$

Rewrite the expression.

${(x + 6)}^{\frac{1}{5} \cdot 5} = 8^{\frac{5}{3}}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(x + 6)}^{\frac{1}{5} \cdot 5} = 8^{\frac{5}{3}}$

Rewrite the expression.

${(x + 6)}^{1} = 8^{\frac{5}{3}}$

Simplify.

$x + 6 = 8^{\frac{5}{3}}$

Simplify the right side.

Simplify $8^{\frac{5}{3}}$ .

Simplify the expression.

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

$x + 6 = {(2^{3})}^{\frac{5}{3}}$

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

$x + 6 = 2^{3 (\frac{5}{3})}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x + 6 = 2^{3 (\frac{5}{3})}$

Rewrite the expression.

$x + 6 = 2^{5}$

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

$x + 6 = 32$

Step 10

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

Subtract $6$ from both sides of the equation.

$x = 32 - 6$

Subtract $6$ from $32$ .

$x = 26$

Solve by Factoring (x+6)^(3/5)=8

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