Solve by Factoring 3(x+3)^(3/4)=81

$3 {(x + 3)}^{\frac{3}{4}} = 81$

Step 1

Subtract $81$ from both sides of the equation.

$3 {(x + 3)}^{\frac{3}{4}} - 81 = 0$

Step 2

Factor $3$ out of $3 {(x + 3)}^{\frac{3}{4}} - 81$ .

Factor $3$ out of $3 {(x + 3)}^{\frac{3}{4}}$ .

$3 ({(x + 3)}^{\frac{3}{4}}) - 81 = 0$

Factor $3$ out of $- 81$ .

$3 {(x + 3)}^{\frac{3}{4}} + 3 \cdot - 27 = 0$

Factor $3$ out of $3 {(x + 3)}^{\frac{3}{4}} + 3 \cdot - 27$ .

$3 ({(x + 3)}^{\frac{3}{4}} - 27) = 0$

Step 3

Rewrite ${(x + 3)}^{\frac{3}{4}}$ as ${({(x + 3)}^{\frac{1}{4}})}^{3}$ .

$3 ({({(x + 3)}^{\frac{1}{4}})}^{3} - 27) = 0$

Step 4

Rewrite $27$ as $3^{3}$ .

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

Step 5

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 $a = {(x + 3)}^{\frac{1}{4}}$ and $b = 3$ .

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

Step 6

Factor.

Simplify.

Multiply the exponents in ${({(x + 3)}^{\frac{1}{4}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move $3$ to the left of ${(x + 3)}^{\frac{1}{4}}$ .

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

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

$3 (({(x + 3)}^{\frac{1}{4}} - 3) ({(x + 3)}^{\frac{1}{2}} + 3 {(x + 3)}^{\frac{1}{4}} + 9)) = 0$

Reorder terms.

$3 (({(x + 3)}^{\frac{1}{4}} - 3) (3 {(x + 3)}^{\frac{1}{4}} + {(x + 3)}^{\frac{1}{2}} + 9)) = 0$

Remove unnecessary parentheses.

$3 ({(x + 3)}^{\frac{1}{4}} - 3) (3 {(x + 3)}^{\frac{1}{4}} + {(x + 3)}^{\frac{1}{2}} + 9) = 0$

Step 7

Divide each term in $3 ({(x + 3)}^{\frac{1}{4}} - 3) (3 {(x + 3)}^{\frac{1}{4}} + {(x + 3)}^{\frac{1}{2}} + 9) = 0$ by $3$ and simplify.

Divide each term in $3 ({(x + 3)}^{\frac{1}{4}} - 3) (3 {(x + 3)}^{\frac{1}{4}} + {(x + 3)}^{\frac{1}{2}} + 9) = 0$ by $3$ .

$\frac{3 ({(x + 3)}^{\frac{1}{4}} - 3) (3 {(x + 3)}^{\frac{1}{4}} + {(x + 3)}^{\frac{1}{2}} + 9)}{3} = \frac{0}{3}$

Simplify the left side.

Reduce the expression by cancelling the common factors.

Cancel the common factor.

$\frac{3 ({(x + 3)}^{\frac{1}{4}} - 3) (3 {(x + 3)}^{\frac{1}{4}} + {(x + 3)}^{\frac{1}{2}} + 9)}{3} = \frac{0}{3}$

Divide $({(x + 3)}^{\frac{1}{4}} - 3) (3 {(x + 3)}^{\frac{1}{4}} + {(x + 3)}^{\frac{1}{2}} + 9)$ by $1$ .

$({(x + 3)}^{\frac{1}{4}} - 3) (3 {(x + 3)}^{\frac{1}{4}} + {(x + 3)}^{\frac{1}{2}} + 9) = \frac{0}{3}$

Expand $({(x + 3)}^{\frac{1}{4}} - 3) (3 {(x + 3)}^{\frac{1}{4}} + {(x + 3)}^{\frac{1}{2}} + 9)$ by multiplying each term in the first expression by each term in the second expression.

${(x + 3)}^{\frac{1}{4}} (3 {(x + 3)}^{\frac{1}{4}}) + {(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 {(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{4}} + {(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Multiply ${(x + 3)}^{\frac{1}{4}}$ by ${(x + 3)}^{\frac{1}{4}}$ by adding the exponents.

Move ${(x + 3)}^{\frac{1}{4}}$ .

$3 ({(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{4}}) + {(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

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

$3 {(x + 3)}^{\frac{1}{4} + \frac{1}{4}} + {(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Combine the numerators over the common denominator.

$3 {(x + 3)}^{\frac{1 + 1}{4}} + {(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Add $1$ and $1$ .

$3 {(x + 3)}^{\frac{2}{4}} + {(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$3 {(x + 3)}^{\frac{2 (1)}{4}} + {(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Cancel the common factors.

Factor $2$ out of $4$ .

$3 {(x + 3)}^{\frac{2 \cdot 1}{2 \cdot 2}} + {(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Cancel the common factor.

Rewrite the expression.

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Multiply ${(x + 3)}^{\frac{1}{4}}$ by ${(x + 3)}^{\frac{1}{2}}$ by adding the exponents.

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

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4} + \frac{1}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4} + \frac{2}{2 \cdot 2}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Multiply $2$ by $2$ .

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1}{4} + \frac{2}{4}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Combine the numerators over the common denominator.

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{1 + 2}{4}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Add $1$ and $2$ .

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{3}{4}} + {(x + 3)}^{\frac{1}{4}} \cdot 9 - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Move $9$ to the left of ${(x + 3)}^{\frac{1}{4}}$ .

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{3}{4}} + 9 \cdot {(x + 3)}^{\frac{1}{4}} - 3 (3 {(x + 3)}^{\frac{1}{4}}) - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Multiply $3$ by $- 3$ .

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{3}{4}} + 9 {(x + 3)}^{\frac{1}{4}} - 9 {(x + 3)}^{\frac{1}{4}} - 3 {(x + 3)}^{\frac{1}{2}} - 3 \cdot 9 = \frac{0}{3}$

Multiply $- 3$ by $9$ .

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{3}{4}} + 9 {(x + 3)}^{\frac{1}{4}} - 9 {(x + 3)}^{\frac{1}{4}} - 3 {(x + 3)}^{\frac{1}{2}} - 27 = \frac{0}{3}$

Combine the opposite terms in $3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{3}{4}} + 9 {(x + 3)}^{\frac{1}{4}} - 9 {(x + 3)}^{\frac{1}{4}} - 3 {(x + 3)}^{\frac{1}{2}} - 27$ .

Subtract $9 {(x + 3)}^{\frac{1}{4}}$ from $9 {(x + 3)}^{\frac{1}{4}}$ .

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{3}{4}} + 0 - 3 {(x + 3)}^{\frac{1}{2}} - 27 = \frac{0}{3}$

Add $3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{3}{4}}$ and $0$ .

$3 {(x + 3)}^{\frac{1}{2}} + {(x + 3)}^{\frac{3}{4}} - 3 {(x + 3)}^{\frac{1}{2}} - 27 = \frac{0}{3}$

Subtract $3 {(x + 3)}^{\frac{1}{2}}$ from $3 {(x + 3)}^{\frac{1}{2}}$ .

$0 + {(x + 3)}^{\frac{3}{4}} - 27 = \frac{0}{3}$

Add $0$ and ${(x + 3)}^{\frac{3}{4}}$ .

${(x + 3)}^{\frac{3}{4}} - 27 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

${(x + 3)}^{\frac{3}{4}} - 27 = 0$

Step 8

Add $27$ to both sides of the equation.

${(x + 3)}^{\frac{3}{4}} = 27$

Step 9

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

${({(x + 3)}^{\frac{3}{4}})}^{\frac{4}{3}} = 27^{\frac{4}{3}}$

Step 10

Simplify the exponent.

Simplify the left side.

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

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

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

${(x + 3)}^{\frac{3}{4} \cdot \frac{4}{3}} = 27^{\frac{4}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x + 3)}^{\frac{3}{4} \cdot \frac{4}{3}} = 27^{\frac{4}{3}}$

Rewrite the expression.

${(x + 3)}^{\frac{1}{4} \cdot 4} = 27^{\frac{4}{3}}$

Cancel the common factor of $4$ .

Cancel the common factor.

${(x + 3)}^{\frac{1}{4} \cdot 4} = 27^{\frac{4}{3}}$

Rewrite the expression.

${(x + 3)}^{1} = 27^{\frac{4}{3}}$

Simplify.

$x + 3 = 27^{\frac{4}{3}}$

Simplify the right side.

Simplify $27^{\frac{4}{3}}$ .

Simplify the expression.

Rewrite $27$ as $3^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$x + 3 = 3^{4}$

Raise $3$ to the power of $4$ .

$x + 3 = 81$

Step 11

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

Subtract $3$ from both sides of the equation.

$x = 81 - 3$

Subtract $3$ from $81$ .

$x = 78$

Solve by Factoring 3(x+3)^(3/4)=81

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